Equivariant eta forms and equivariant differential $ K $-theory

arXiv:1610.02311v1 [math.DG] 7 Oct 2016

EQUIVARIANT ETA FORMS AND EQUIVARIANT DIFFERENTIAL K-THEORY BO LIU Abstract. In this paper, we construct an analytic model of equivariant differential K-theory for closed manifolds with almost free action, which is a ring valued functor with the usual properties of a differential extensions of a cohomology theory, using the equivariant BismutCheeger eta forms with the equivariant version of spectral sections developed by MelrosePiazza. In fact, it could also be regarded as an analytic model of differential K-theory for closed orbifolds. Furthermore, we construct a well-defined push-forward map in equivariant differential K-theory and prove the properties of it. In order to do these, we extend the Melrose-Piazza spectral section to the equivariant case, introduce the equivariant version of higher spectral flow for arbitrary dimensional fibers and use them to prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms.

0. Introduction By the de Rham theory, the de Rham cohomology of a smooth manifold can be represented by differential forms, thus getting the global information by means of local data. In a similar way, a generalized differential cohomology theory gives a way to combine the cohomological information with differential geometric objects. An important case is the differential K-theory. The differential K-theory is partly motivated by the study of D-branes in theoretical physics introduced by Witten [36] in 1998. He points out that D-branes carry Ramond-Ramond charges that massless Ramond-Ramond fields are differential forms, and that the D-brane charges should be understood in terms of K-theory. Various models of differential K-theory for manifolds have been proposed (see e.g. Bunke-Schick [12], Freed-Lott [19], Hopkins-Singer [20] and Simons-Sullivan [34]). We note that in the model of Bunke-Schick [12], the differential K-theory is constructed using the geometric families and Bismut-Cheeger eta forms with taming. Until now, the equivariant version of the differential K-theory is not well understood yet. When the group action is finite, the equivariant differential K-theory is studied by SzaboValentino [35] and Ortiz [30]. In [14], Bunke and Schick extend their model to the orbifold case using the language of stacks, which could be regarded as a model of equivariant differential K-theory for the almost free action. Inspired by the model of Bunke and Schick [14], as a parallel version, in this paper, we construct a purely analytic model of equivariant differential K-theory for closed manifolds with almost free action using the local index technique developed by [9]. Furthermore, we prove that the push-forward map is well-defined in our construction, which is a problem proposed in [12, 14] and the main motivation for this new construction. This model is a direct generalization of [12] without using the language of stacks and could also be regarded as an analytic model of differential K-theory for closed orbifolds. Note that when restricted to the non-equivariant case, our construction is a little different from that in [12] although they are isomorphic. We use the analytic tools: spectral sections and higher spectral flows instead of the taming and Kasporov KK-theory in [12]. 1

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The key tools in our model are the equivariant Bismut-Cheeger eta forms [6] with the equivariant version of Melrose-Piazza spectral sections [28, 29] and Dai-Zhang higher spectral flows [16]. In [28, 29], in order to prove the family index theorem for manifolds with boundary, Melrose and Piazza define the spectral section for a family of self-adjoint (resp. odd Z2 -graded self-adjoint) 1st-order elliptic pseudodifferential operators on a family of odd (resp. even) dimensional manifolds when the family index of the operators vanishes. In [16], using the spectral section, Dai and Zhang introduce the higher spectral flow for a family of Dirac type operators on a family of odd dimensional manifolds. In this paper, in order to construct the analytic model of the equivariant differential Ktheory, we extend the spectral section, the higher spectral flow and the eta form to the equivariant case. Especially, we introduce the equivariant higher spectral flow for a family of even dimensional manifolds. Furthermore, we prove the anomaly formula and the functoriality of equivariant eta forms using the language of equivariant higher spectral flow, which is an analogue of the results in [11, 23] and using the techniques in [15, 24, 25, 26]. Note that the proof of the funtoriality of equivariant eta forms is highly nontrivial. It is a question proposed in [14]. Let π : W → B be a proper submersion of smooth closed manifolds with oriented fibers Z. Let T Z = T W/B be the relative tangent bundle to the fibers Z with Riemannian metric gT Z and T H W be a horizontal subbundle of T W , such that T W = T H W ⊕ T Z. Let o ∈ o(T Z) be an orientation of T Z. Let ∇T Z be the Euclidean connection on T Z defined in (1.7). We assume that T Z has a Spinc structure. Let LZ be the complex line bundle associated with the Spinc structure of T Z with a Hermitian metric hLZ and a Hermitian connection ∇LZ . Let (E, hE ) be a Z2 -graded Hermitian vector bundle with a Hermitian connection ∇E . Let G be a compact Lie group which acts on W such that π ◦ g = g ◦ π for any g ∈ G. We assume that the G-action preserves everything. We denote by the family of G-equivariant geometric data F = (W, LZ , E, o, T H W, gT Z , hLZ , ∇LZ , hE , ∇E )) an equivariant geometric family. Let D(F) ∗ (B), be the fiberwise Dirac operators of F. Then we have the family index map ind(D(F)) ∈ KG where ∗ = 0 or 1 corresponds to the dimensions of fibers Z are even or odd. Let F0G (B) (resp. F1G (B)) be the set of equivalent classes of isomorphic equivariant geometric families such that the dimension of all fibers are even (resp. odd). We denote by F ∼ F ′ if ind(D(F)) = ind(D(F ′ )). The following proposition is proved in [14]. It could be regarded as a geometric version of the equivariant Atiyah-Jänich theorem. Proposition 0.1. We have the ring isomorphism (0.1)

∗ F∗G (B)/ ∼ ≃ KG (B).

Let D be a family of 1st-order pseudodifferential operator on F along the fibers, which is self-adjoint, fiberwisely elliptic and commutes with the G-action and the principal symbol of which is the same as that of D(F). Furthermore, if F is even (resp. odd), we assume that D anti-commutes (resp. commutes) with the Z2 -grading. As in [16], we call such D is an ∗ (B) and at least one equivariant B-family on F (see Definition 2.1). If ind(D) = 0 ∈ KG component of the fiber has nonzero dimension, there exists an equivariant spectral section P and a smooth operator AP associated with P , such that D + AP is an invertible equivariant B-family (see Proposition 2.3). Let P , Q be equivariant spectral sections, we could define the ∗ (B) (see (2.5) and (2.11)). difference [P − Q] ∈ KG Let F, F ′ ∈ F1G (B) (resp. F0G (B)) which have the same topological structure, that is, the only differences between them are metrics and connections. Let D0 , D1 be two equivariant

EQUIVARIANT ETA FORMS AND EQUIVARIANT DIFFERENTIAL K-THEORY

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B-families on F, F ′ respectively. Let Q0 , Q1 be equivariant spectral sections with respect to D0 , D1 respectively. We define the equivariant higher spectral flow sf G {(D0 , Q0 ), (D1 , Q1 )} 0 (B) (resp. K 1 (B)) in Definition between the pairs (D0 , Q0 ), (D1 , Q1 ) to be an element in KG G 2.5 and 2.6. Note that when F is odd, it is the direct extension of higher spectral flow in [16]; when F is even, it is defined by adding an additional dimension. Moreover, besides the equivariant geometric family, we could also represent the element of equivariant K-group as equivariant higher spectral flow. 0 (B) (resp. K 1 (B)), there exists F ∈ F1 (B) (resp. F0 (B)) Proposition 0.2. For any x ∈ KG G G G and equivariant spectral sections P , Q with respect to D(F), such that [P − Q] = x.

From this point of view, the equivariant higher spectral flow here is the same as the term ind((E ×I)bt ) in [14, 2.5.8], which uses the KK-theory. This enable us to replace the techniques of KK-theory in [14] by that of equivariant higher spectral flow, which is purely analytic. Let D be an equivariant B-family on F. A perturbation operator with respect to D is a family of bounded pseudodifferential operators A such that D + A is an invertible equivariant B-family on F, which is a generalization of AP . Note that if at least one component of the fibers of F has nonzero dimension, a perturbation ∗ (B). operator exists with respect to D if and only if ind(D(F)) = 0 ∈ KG If the G-action on B is trivial, for any g ∈ G, we can define an equivariant eta form η˜g (F, A) with respect to a perturbation operator A in Definition 2.11. If the equivariant geometric families F and F ′ have the same topological structure, we prove the anomaly formula as follows. Theorem 0.3. Assume that the G-action on B is trivial. Let F, F ′ ∈ F∗G (B) which have the same topological structure. Let A, A′ be perturbation operators with respect to D(F), D(F ′ ) and P , P ′ be the APS projections (see Proposition 2.3) with respect to D(F) + A, D(F ′ ) + A′ respectively. For any g ∈ G, modulo exact forms, we have Z

f g (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) ∧ chg (E, ∇E ) Td (0.2) η˜g (F , A ) − η˜g (F, A) = Zg Z ′ ′ e g (∇E , ∇′ E ) + chg sf G {(D(F) + A, P ), (D(F ′ ) + A′ , P ′ )} , Tdg (∇ T Y , ∇ LY ) ∧ ch + ′

′

Zg

where Z g is the fixed point set of g on the fibers Z and the characteristic forms and ChernSimons forms are defined in Section 2. Note that when F, F ′ ∈ F0G (B), the proof of the anomaly formula uses a special case of functoriality of equivariant eta forms which is highly nontrivial. Let π : V → B be an equivariant proper submersion with closed oriented equivariant Spinc fibers Y . We assume that V is closed and G acts trivially on B. Then an equivariant geometric family FX over V induces an equivariant geometric family FZ over B (see (1.25)). For any g ∈ G, let Y g and Z g be the fixed point sets of g on the fibers Y and Z respectively. We obtain the functoriality of equivariant eta forms.

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Theorem 0.4. Let AZ and AX be perturbation operators with respect to D(FZ ) and D(FX ). Then for T ≥ 1 large enough and any g ∈ G, modulo exact forms, we have Z Tdg (∇T Y , ∇LY ) ∧ ηeg (FX , AX ) (0.3) ηeg (FZ , AZ ) = g Y Z f g (∇T Z , ∇LZ , ∇T Y,T X , ∇LZ ) ∧ chg (E, ∇E ) Td − Zg

b AX , P ′ )}), + chg (sf G {(D(FZ ) + AZ , P ), (D(FZ,T ) + 1⊗T

where FZ,T is the equivariant geometric family defined in (2.68), ∇T Y,T X is defined in (2.69) and P , P ′ are the associated APS projections respectively. In the last section, inspired by [12, 14, 30], we use the results above to define the equivariant differential K-theory for the closed manifolds with the almost free action and study the properties of it. ∗ (B) ⊗ R is Essential to our definition is that when the group action is almost free, KG ∗ (B, R) defined in (3.7), which is the cohoisomorphic to the delocalized cohomology Hdeloc,G ∗ mology of complex (Ωdeloc,G (B, R), d) of differential forms on the disjoint union of the fixed point set of a representative element in the conjugacy classes. Furthermore, we could define ηeG (F, A) ∈ Ω∗deloc,G (B, R)/Imd when the group action is almost free on B. A cycle for an equivariant differential K-theory class over B is a pair (F, ρ), where F ∈ F∗G (B) and ρ ∈ Ω∗deloc,G (B, R)/Im d. The cycle (F, ρ) is called even (resp. odd) if F is even (resp. even odd) and ρ ∈ Ωodd deloc,G (B, R)/Im d (resp. ρ ∈ Ωdeloc,G (B, R)/Im d). Two cycles (F, ρ) and c 0G (B) (resp. (F ′ , ρ′ ) are called isomorphic if F and F ′ are isomorphic and ρ = ρ′ . Let IC c 1 (B)) denote the set of isomorphism classes of even (resp. odd) cycles over B. Let F op IC G be the equivariant geometric family reversing the Z2 -grading of E in F, which implies that ind(D(F op )) = − ind(D(F)). We call two cycles (F, ρ) and (F ′ , ρ′ ) paired if (0.4)

ind(D(F)) = ind(D(F ′ )),

and there exists a perturbation operator A with respect to D(F + F (0.5)

′

ρ − ρ = ηeG (F + F

′ op

′ op

) such that

, A).

Let ∼ denote the equivalence relation generated by the relation ”paired”.

b 0 (B) (resp. K b 1 (B)) is the group Definition 0.5. The equivariant differential K-theory K G G 1 0 c G (B)/ ∼). c G (B)/ ∼ (resp. IC completion of the abelian semigroup IC

Let πY : V → B be an equivariant submersion of closed smooth G-manifolds with closed Spinc fiber Y . We assume that the G-action on B is almost free. Thus, so is the action on V . As in [12], we define the equivariant differential K-orientation with respect to πY in Definition c ∗G (B) in (3.22). Then we prove that c ∗G (V ) → IC 3.7 and the map π bY ! : IC b ∗ (V ) → K b ∗ (B) is well-defined. Theorem 0.6. The map π bY ! : K G G

By Theorem 0.3 and 0.4, in Section 3, we also prove that our model is a ring valued functor with the usual properties of a differential extensions of a cohomology. Finally, we explain that this model could be naturally regarded as a model of differential K-theory for orbifolds. Note that there is no adiabatic limit in Theorem 0.4. So in non-equivariant case, our proofs of these properties simplify that in [12]. This paper is organized as follows.


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In Section 1, we give a geometric description of equivariant K-theory. In Section 2, we extend the Melrose-Piazza spectral section to the equivariant case, introduce the equivariant Dai-Zhang higher spectral flow for arbitrary dimensional fibers and use them to obtain the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms. In Section 3, we construct an analytic model for equivariant differential K-theory and prove some properties. To simplify the notations, we use the Einstein summation convention in this paper. In the whole paper, we use the superconnection formalism of Quillen [32]. If A is a Z2 -graded algebra, and if a, b ∈ A, then we will note [a, b] as the supercommutator of a, b. If B is another b as the Z2 -graded tensor product. Z2 -graded algebra, we will note A⊗B If E = E+ ⊕ E− is a Z2 -graded space, we denote by Trs [A] = Tr |E+ [A] − Tr |E− [A].

(0.6)

For a fiber bundle π : W → B, we will often use the integration of the differential forms along the fibers Z in this paper. Since the fibers may be odd dimensional, we must make precise our sign conventions. If α is a differential form on W which in local coordinates is given by α = dy p1 ∧ · · · ∧ dy pq ∧ β(x)dx1 ∧ · · · ∧ dxn ,

(0.7) we set (0.8)

Z

α = dy Z

p1

∧ · · · ∧ dy

pq

Z

Z

β(x)dx1 ∧ · · · ∧ dxn .

1. Equivariant K-theory In this section, we give a geometric description of equivariant K-theory for compact Lie groups, which could be regarded as an analogue of de Rham theory, and define the pushforward map of it. The setting in this section is the equivariant extension of those in [12]. 1.1. Clifford algebra. Let V n be an oriented Euclidean space, such that dim V n = n, with orthonormal basis {ei }1≤i≤n . Let C(V n ) be the complex Clifford algebra of V n defined by the relations (1.1)

ei ej + ej ei = −2δij .

To avoid ambiguity, we denote by c(ei ) the element of C(E) corresponding to ei . We consider the group Spincn as a multiplicative subgroup of the group of units of C(E). For the definition and the properties of the group Spincn , see [21, Appendix D]. For n = 2k, even, up to isomorphism, C(V n ) has a unique irreducible module, Sn , which has dimension 2k and is Z2 -graded. Let τV be the Z2 -grading of V . If n = 2k − 1 is odd, C(V n ) has two inequivalent irreducible modules, each of dimension 2k−1 . However, they are equivalent when restricted to Spincn . We still denote it by Sn . Let W m be another real inner product space with orthonormal basis {fp }. Then as Clifford algebras, (1.2)

n b C(W m ⊕ V n ) ≃ C(W m )⊗C(V ).

Let SW and SV be the spinors respectively. b V is SW ⊗SV If either m or n is even, we simply assume m is even, the spinor SW ⊕V = SW ⊗S b := c(fp ) ⊗ 1, 1⊗c(e b i ) := τW ⊗ c(ei ). with Clifford action c(fp )⊗1 b V is SW ⊗ SV ⊗ C2 . Let If m and n are all odd, the spinor SW ⊕V = SW ⊗S √ −1 0 1 0 √ . Γ1 = , Γ2 = 0 − −1 1 0

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b := c(fp ) ⊗ 1 ⊗ Γ1 , 1⊗c(e b i ) := 1 ⊗ c(ei ) ⊗ Γ2 . The The Clifford actions are defined by c(fp )⊗1 Z2 -grading of SW ⊕V is 1 0 τW ⊕V := IdSW ⊗ IdSV ⊗ . 0 −1 1.2. Equivariant geometric family. In this subsection, we introduce the equivariant geometric family. Let π : W → B be a smooth proper submersion of smooth closed manifolds with closed fibers Z (maybe non-connected). Let B = ⊔i Bi be a finite disjoint union of closed connected manifolds. Let Wi be the restriction of W on Bi . Let Wi = ⊔j Wij be a finite disjoint union of closed connected manifolds. Let Zij be the fibers of the submersions restricted on Wij . We note here that the dimension of Zij might be zero. In the sequel, we will often omit the subscript i, j. Let T Z = T W/B be the relative tangent bundle to the fibers Z. We assume that T Z is orientable with an orientation o ∈ o(T Z). Let TπH W be a horizontal subbundle of T W such that (1.3)

T W = TπH W ⊕ T Z.

The splitting (1.3) gives an identification (1.4)

TπH W ∼ = π ∗ T B.

If there is no ambiguity, we will omit the subscript π in TπH W . Let P T Z be the projection (1.5)

P T Z : T W = T H W ⊕ T Z → T Z.

Let gT Z , gT B be Riemannian metrics on T Z, T B. We equip T W = T H W ⊕ T Z with the Riemannian metric (1.6)

gT W = π∗ gT B ⊕ gT Z .

Let ∇T W , ∇T B be the Levi-Civita connections on (W, g T W ), (B, gT B ). Set (1.7)

∇T Z = P T Z ∇T W P T Z .

Then ∇T Z is a Euclidean connection on T Z. By [5, Theorem 1.9], we know that ∇T Z only depends on (T H W, gT Z ). Let C(T Z) be the Clifford algebra bundle of (T Z, gT Z ), whose fiber at x ∈ W is the Clifford algebra C(Tx Z) of the Euclidean space (Tx Z, gTx Z ). We make the assumption that T Z has a Spinc structure. Then there exists a complex line bundle LZ over W such that ω2 (T Z) = c1 (LZ ) mod (2). Let S(T Z, LZ ) be the fundamental complex spinor bundle for (T Z, LZ ), which has a smooth action of C(T Z) (cf. [21, Appendix D.9]). Locally, the spinor S(T Z, LZ ) may be written as (1.8)

1/2

S(T Z, LZ ) = S(T Z) ⊗ LZ ,

where S(T Z) is the fundamental spinor bundle for the (possibly non-existent) spin structure on 1/2 T Z, and LZ is the (possibly non-existent) square root of LZ . Let hLZ be the Hermitian metric on LZ and ∇LZ be the Hermitian connection on (LZ , hLZ ). Let hSZ be the Hermitian metric on S(T Z, LZ ) induced by gT Z and hLZ and ∇SZ be the connection on S(T Z, LZ ) induced by ∇T Z


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and ∇LZ from (1.8). Then ∇SZ is a Hermitian connection on (S(T Z, LZ ), hSZ ). Moreover, it is a Clifford connection associated with ∇T Z , i.e., for any U ∈ T W , V ∈ C ∞ (W, T Z), i h (1.9) ∇SUZ , c(V ) = c ∇TU Z V .

If n = dim Z is even, the spinor S(T Z, LZ ) is Z2 -graded and the action of T Z exchanges the Z2 -grading. In the following, we often simply denote the spinor by SZ . Let E = E+ ⊕ E− be a Z2 -graded Hermitian vector bundle over W with Hermitian metric E h , for which E+ and E− are orthogonal, and ∇E a Hermitian connection on (E, hE ). Set (1.10)

b b + 1⊗∇ b E. ∇SZ ⊗E := ∇SZ ⊗1

b b hSZ ⊗ hE ). is a Hermitian connection on (SZ ⊗E, Then ∇SZ ⊗E Let G be a compact Lie group which acts on W such that for any g ∈ G, π ◦ g = g ◦ π. We assume that the action of G preserves the splitting (1.3), the Spinc structure of T Z and gT Z , hLZ , ∇LZ are G-invariant. We assume that E is a G-equivariant Z2 -graded complex vector bundle and hE , ∇E are G-invariant.

Definition 1.1. An equivariant geometric family F over B is a family of G-equivariant geometric data (1.11)

F = (W, LZ , E, o, T H W, gT Z , hLZ , ∇LZ , hE , ∇E )

described as above. We call the equivariant geometric family F is even (resp. odd) if for any connected component of fibers, the dimension of it is even (resp. odd). Definition 1.2. Let F and F ′ be two equivariant geometric families over B. An isomorphism ∼ F → F ′ consists of the following data: F

b SZ ⊗E

b ′ SZ ′ ⊗E

f

W

π

W′

π

′

B where 1. f is a diffeomorphism commuting with the G-action such that π ′ ◦ f = π, 2. f preserves the G-invariant orientation and Spinc structure of the relative tangent bundle, 3. F is an equvariant bundle isomorphism over f preserving the grading of the vector bundle and the spinor (if it has), 4. f preserves the horizontal subbundle and the vertical metric, 5. F preserves the metrics and the connections. If only the first three conditions hold, we say that F and F ′ have the same topological structure. Let F0G (B) (resp. F1G (B)) be the set of equivalent classes of even (resp. odd) equivariant geometric families. For two equivariant geometric families F, F ′ , we can form their sum F + F ′ over B as a new equivariant geometric family. The underlying fibration with closed fibers of the sum is π ⊔ π ′ : W ⊔ W ′ → B, where ⊔ is the disjoint union. The remaining structures of F + F ′ are

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induced in the obvious way. Let FG∗ (B) = FG0 (B) ⊕ FG1 (B) be the set of equivalent classes of equivariant geometric families. It is a additive semigroup. Let f : B × B ′ → B be the projection onto the first part. For any F ∈ F∗G (B), we could construct the pullback f ∗ F in a natural way. Definition 1.3. The opposite family F op of an equivariant geometric family F is obtained by reversing the Z2 -grading of E. 1.3. Equivariant K-Theory. In this subsection, we give some examples of the equivariant geometric families and get a geometric description of equivariant K-theory. 0 (B) is the Grothendieck group of the equivalent Recall that [33] the equivariant K-group KG classes of equivariant vector bundles over B. Let i : B → B × S 1 be a G-equivariant inclusion map. It is well known that if the G-action on S 1 is trivial, 1 0 0 (1.12) KG (B) ≃ ker i∗ : KG (B × S 1 ) → KG (B) . Let {ei } be a local orthonormal frame of T Z. Let D(F) be the fiberwise Dirac operator

(1.13)

b

D(F) = c(ei )∇SeiZ ⊗E

associated with F ∈ F∗G (B). Then the G-action commutes with D(F). Thus the classical construction of Atiyah-Singer assigns to this family its equivariant (analytic) index ind(D(F)) ∈ 0 (B) (resp. K 1 (B)) when F ∈ F0 (B) (resp. F1 (B)) [2, 3]. KG G G G ∗ (B) = K 0 (B) ⊕ K 1 (B). Since Let KG G G (1.14)

∗ ind(D(F + F ′ )) = ind(D(F)) + ind(D(F ′ )) ∈ KG (B),

the equivariant index defines a semigroup homomorphism (1.15)

∗ ind : F∗G (B) → KG (B),

F

7→ ind(D(F)).

Furthermore, for F, F ′ ∈ F∗G (B), we can form their cup product F ∪ F ′ over B. The underlying fibration with closed fibers of the cup product is π ∪ π ′ : W ×B W ′ → B. The vector b Z ′ is constructed as in Section 1.1. The b ⊗E b ′ . Here the spinor SZ ⊗S b Z ′ )⊗E bundle now is (SZ ⊗S ′ remaining structures of F ∪ F are induced in the obvious way. It is well-known that (1.16)

∗ ind(D(F ∪ F ′ )) = ind(D(F)) ∪ ind(D(F ′ )) ∈ KG (B).

Example 1.4. a) Let (E, hE ) be an equivariant Z2 -graded Hermitian vector bundle over B with a G-invariant Hermitian connection ∇E . Then (E, hE , ∇E ) can be regarded as an even equivariant geometric family F for Z = pt. In this case, D(F) = 0 and ind(D(F)) = [E+ ] − 0 (B). [E− ] ∈ KG b) We choose F as in a). Let W ′ = B × S 2 and γ be the canonical nontrivial complex line bundle on S 2 = CP 1 . Then γ can be naturally extended on W ′ . Thus (W ′ , γ) with canonical metrics, connections and the standard orientation o ∈ o(S 2 ) forms an even geometric family F ′ over B. In this case, since ind(DSγ 2 ) = 1, where DSγ 2 is the Dirac operator on S 2 associated 0 (B). with γ, from (1.16), we could get that ind(D(F ∪ F ′ )) = ind(D(F)) ∈ KG c) Let B = S 1 = R/Z, W = S 1 × S 1 and π be the natural projection on the first part. We consider the Hermitian line bundle (L, hL ) which is obtained by identifying (θ = 0, t, v) ∈ √ [0, 1] × S 1 × C and (θ = 1, t, exp(−2πt −1)v) ∈ [0, 1] × S 1 × C. The line bundle L is naturally √ endowed a Hermitian connection ∇L = d + 2π(θ − 1/2) −1dt. We choose the Z2 -grading of L such that L+ = L and L− = 0. Then we get an odd geometric family F L after choosing the


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natural geometric data (oL ∈ o(T S 1 ), TπH (S 1 × S 1 ), gT S ). In fact, ind(D(F L )) is a generator of K 1 (S 1 ) ≃ Z. d) Let F ∈ F∗G (B). Let p1 and p2 be the natural projection onto the first and second part of B × S 1 respectively. We choose F L as in c). Then p∗1 F ∪ p∗2 F L is an equivariant geometric family over B × S 1 . From the proof of [8, Theorem 2.10], for F ∈ F1G (B), there exists inclusion 1 (B) in the i : B → B × S 1 such that i∗ ind(D(p∗1 F ∪ p∗2 F L )) = 0. Moreover, as a element of KG sense of (1.12), by [29, Proposition 6], we have (1.17)

ind(D(F)) = ind(D(p∗1 F ∪ p∗2 F L )).

In fact, we could prove (1.16) by (1.17). This example is essential in our construction of higher spectral flow for even case. We denote by F ∼ F ′ if ind(D(F)) = ind(D(F ′ )). It is an equivalent relation and compatible with the semigroup structure. So F∗G (B)/ ∼ is a semigroup and the map (1.18)

∗ ind : F∗G (B)/ ∼ −→ KG (B)

is an injective semigroup homomorphism. By Definition 1.3, we have (1.19)

ind(D(F op )) = − ind(D(F)).

After defining −F := F op , the semigroup F∗G (B)/ ∼ can be regarded as an abelian group. So, by (1.19), the equivariant index map in (1.18) is a group homomorphism. Furthermore, by (1.16), the equivariant index map in (1.18) is a ring homomorphism. The following proposition is proved in [14]. We sketch the proof for completion. Proposition 1.5. The equivariant index map ind in (1.18) is surjective. In other words, we have the Z2 -graded ring isomorphism (1.20)

∗ F∗G (B)/ ∼ ≃ KG (B).

Proof. When ∗ = 0, we can get the proposition directly from Example 1.4 a) or b). 1 (B), When ∗ = 1, by [31, Theorem 2.8.8] (see also [14, Section 2.5.8]), for any [F ] ∈ KG there exists a finite dimensional G-representation V , such that [F ] can be represented as a G-invariant unitary element of End(V ) ⊗ C 0 (B, C). For (b, t, v) ∈ B × [0, 1] × V , the relation (b, 0, v) ∼ (b, 1, F (b)v) makes a G-equivariant vector bundle W over B ×S 1 . Let U = B ×S 1 ×V 0 (B ×S 1 ) corresponds to [F ] ∈ K 1 (B) be the G-equivariant trivial bundle. Then [W ]−[U ] ∈ KG G under the isomorphism (1.12). Moreover, after taking the natural geometric data, we get an odd equivariant geometric family F over B with fiber S 1 and Z2 -graded equivariant vector bundle W ⊕ U . As in [10, Section 4.2.3], we can get ind(D(F)) = [F ]. The proof of Proposition 1.5 is complete. Remark 1.6. Note that if we replace the Spinc condition of the geometric family by the general Clifford module condition (which is the setting in [12]) or the Spin condition, Proposition 1.5 also holds. Since we don’t use the language of Clifford modules here, our definition of F op in Definition 1.3 is simper than that in [12]. In fact, in the sense of (1.20), they are the same. 1.4. Push forward map. In this subsection, we define the push-forward map of equivariant K-theory using the equivariant geometric families. Let πY : V → B be a G-equivariant submersion of smooth closed manifolds with closed Spinc fibers Y .

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Definition 1.7. An equivariant K-orientation of πY is an equivariant Spinc structure of T Y . We call the equivariant K-orientation is even (resp. odd) if for any connected component of 0 (π ) (resp. O 1 (π )) be the set of fibers Y , the dimension of it is even (resp. odd). Let OG Y G Y even (resp. odd) equivariant K-orientations. ∗ (π ), we will use Proposition 1.5 to define If πY has an equivariant K-orientation OY ∈ OG Y ∗′ (V ) → K ∗′ +∗ (B) as follows. the push-forward map of equivariant K-groups πY ! : KG G Let πX : W → V be a G-equivariant submersion of smooth closed manifolds with closed Spinc fibers X and

(1.21)

FX = (W, LX , E, oX , TπHX W, gT X , hLX , ∇LX , hE , ∇E ) ′

be a G-equivariant geometric family in F∗G (V ). Then πZ := πY ◦ πX : W → B is a smooth submersion with closed Spinc fibers Z. Then we have the diagram of smooth fibrations: X

Z

W πZ

πX

Y

πY

V

B.

Set TπHX Z := TπHX W ∩ T Z. Then we have the splitting of smooth vector bundles over W , (1.22)

T Z = TπHX Z ⊕ T X,

and (1.23)

∗ T Y. TπHX Z ∼ = πX

Then we can get a G-invariant orientation oY ∪ oX ∈ o(T Z). Set (1.24)

∗ LY ⊗ LX . LZ := πX

Take the geometric data (TπHZ W, gT Z , hLZ , ∇LZ ) of πZ such that TπHZ W ⊂ TπHX W , gT Z = ∗ g T Y ⊕ g T X , hLZ = π ∗ hLY ⊗ hLX and ∇LZ = π ∗ ∇LY ⊗ 1 + 1 ⊗ ∇LX . We get a new πX X X equivariant geometric family over B (1.25)

FZ := (W, LZ , E, oY ∪ oX , TπHZ W, gT Z , hLZ , ∇LZ , hE , ∇E ).

In the sense of (1.20), we will prove that ∗ (π ) fixed, the push-forward map Theorem 1.8. For equivariant K-orientation OY ∈ OG Y ′

(1.26)

′

∗ +∗ ∗ (B), πY ! : KG (V ) → KG

[FX ]

7→ [FZ ]

is a well-defined group homomorphism and independent of the geometric data (TπHZ W, gT Z , hLZ , ∇LZ ) of πZ . Proof. From the definition and the homotopy invariance property of the equivariant family index, the push-forward map in (1.26) is a group homomorphism and independent of the geometric data (TπHZ W, gT Z , hLZ , ∇LZ ) of πZ . We will prove the well-defined property in Lemma 2.15 later. Let πU : B → S be a G-equivariant submersion of smooth closed manifolds with closed Spinc fibers U and an equivariant K-orientation OU . Then πA := πU ◦ πY : V → S is a Gequivariant submersion with an equivariant K-orientation constructed by OY and OU . From the construction of the push-forward map and Theorem 1.8, the following theorem is obvious.


11

∗ (V ) → K ∗ (S) Theorem 1.9. We have the equality of homomorphisms KG G

πA ! = πU ! ◦ πY !.

(1.27)

2. Equivariant higher spectral flows and equivariant eta forms In this section, we extend the Melrose-Piazza spectral section to the equivariant case, introduce the equivariant version of Dai-Zhang higher spectral flow for arbitrary dimensional fibers and use them to obtain the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms. 2.1. Equivariant spectral section and equivariant higher spectral flow. In this subsection, we define the equivariant version of spectral section and higher spectral flow and give a model of equivariant K-theory using the equivariant higher spectral flow. Definition 2.1. (compare with [16, Definition 1.6]) Let F ∈ F∗G (B). We call an operator D is an equivariant B-family on F if it is a 1st-order pseudodifferential operator on F along the fiber, which is self-adjoint, fiberwisely elliptic and commutes with the G-action, such that (a) its principal symbol is given by that of D(F); (b) it preserves the Z2 -grading of E when the fiber is odd dimensional; b when the fiber is even dimensional. (c) it anti-commutes with the Z2 -grading of SZ ⊗E Note that the fiberwise Dirac operator D(F) is an equivariant B-family. In fact, for any ∗ (B). equivariant B-family D, we have ind(D) = ind(D(F)) ∈ KG

Definition 2.2. (compare with [28, Definition 1] and [29, Definition 1]) The equivariant Melrose-Piazza spectral section is a family of self-adjoint pseudodifferential projections P with respect to an equivariant B-family D, which commutes with the G-action, such that (a) for some smooth function f : B → R (depending on P ) and every b ∈ B, ( Pb u = u, if λ > f (b); Db u = λu =⇒ (2.1) Pb u = 0, if λ < −f (b); (b) if F is odd, P commutes with the Z2 -grading of E; (c) if F is even, (2.2)

τ P + P τ = τ,

b where τ is the Z2 -grading of SZ ⊗E.

The following proposition is the natural equivariant extension of the results in [28, 29].

Proposition 2.3. Let F ∈ F∗G (B) and D be an equivariant B-family on F. (i) (compare with [28, Proposition 1] and [29, Proposition 2]) If there exists an equivariant 0 (B) spectral section with respect to D on F ∈ F0G (B) (resp. F1G (B)), then ind(D) = 0 ∈ KG 1 0 1 (resp. KG (B)). Conversely, If ind(D) = 0 ∈ KG (B) (resp. KG (B)) and at least one component of the fibers has the nonzero dimension, there exists an equivariant spectral section with respect to D. (ii) (compare with [28, Proposition 2]) For F ∈ FG1 (B), given equivariant spectral sections P , Q with respect to D, there exists an equivariant spectral section R with respect to D such that P R = R and QR = R. We say that R majors P , Q. (iii) (compare with [28, Lemma 8] and [29, Lemma 1]) If there is an equivariant spectral section P with respect to D, then there is a family of self-adjoint equivariant smoothing operators

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AP (when the dimension of the fibers are zero, it descends to an endmorphism) with range in a finite sum of eigenspaces of D such that D + AP is an invertible equivariant B-family and P is the Atiyah-Patodi-Singer projection (we often simply denoted by APS projection later) onto the positive part of the spectrum of D + AP . Proof. The proofs of these properties are almost the same as those in [28, 29]. We sketch the proof here. Case 1: Let F ∈ F1G (B). Let Q = D/(1 + D 2 )1/2 . Then Q is bounded, commutes with the G-action and ind(Q) = 1 (B). Since the Hilbert bundle is trivial, we can regard Q as a continuous family ind(D) ∈ KG b of self-adjoint bounded operators {Qb }b∈B on L2 (Z, SZ ⊗E). (i) Assume that there exists an equivariant spectral section P of D. Let K = (1 + D 2 )−1/2 . Then Q is in the same G-homotopy classes of P (Q + rK)P + (1 − P )(Q − rK)(1 − P ) for r > 0. When r is large enough, for any b ∈ B, Pb (Qb +rKb )Pb is positive and (1−Pb )(Qb −rKb )(1−Pb ) 1 (B). is negative. Since ind(Q) is G-homotopy invariant, we have ind(Q) = 0 ∈ KG 1 (B), then Q is G-homotopic to constant. The proof of [28, Proposition If ind(Q) = 0 ∈ KG 1] provides a process to construct a spectral section. All the operators in the process commute with the G-action. (ii) We consider the operators P DP on the range of P . By the definition of the spectral section, there exists N > 0, such that all but the first N eigenfunctions of P DP are eigenfunctions of D. When M > 0 is large enough, using the method in the construction of the equivariant spectral section in the proof of (i) (compare with [22, Proposition 3]), we could find an equivariant subbundle of the range of P contains the first N eigenfunctions and be contained in the span of the first M eigenfunctions. Let R be the orthogonal projection on the complement of the subbundle and extend to be zero on I − P . Then R is an equvariant spectral section such that P R = R. If the integer N is chosen large enough, then the projection R will have range contained in the intersection of the ranges of any two given spectral sections P and Q. So QR = R. (iii) Let Pλ∈[a1 ,a2 ],b (Db ) be the span of the eigenfunctions of Db corresponding to the eigenvalues λ ∈ [a1 , a2 ]. Since B is compact, we can choose s > 0, such that P is an equivariant spectral section for f (b) ≡ s. By the proof of (ii), we can choose equivariant spectral sections Q, R, such that for any b ∈ B, Qb = 0 on Pλ≤s,b (Db ) and Rb = I on Pλ≥−s,b (Db ). Then the operator (2.3)

e = QDQ + sP R(I − Q) + (I − R)D(I − R) − s(I − P )R(I − Q). D

e − D. is an invertible equivariant B-family. We take AP = D 0 Case 2: Let F ∈ FG (B). Since D is odd, the equivariant K 1 index of the whole self-adjoint family D vanishes (see [29, Proposition 2]). Following the same process in the proof of (i) in the odd case, we could obtain that there exists an equivariant spectral section P ′ in the odd sense, which means that it is a equivariant spectral section without the condition (2.2). (iii) By the proof of (ii) for the odd case, if at least one component of the fiber has the nonzero dimension, we could choose P ′ such that P ′ DP ′ is positive on the range of P . Then the operator (2.4)

AP = P − P ′ − τ (P − P ′ )τ + P ′ DP ′ + τ P ′ τ Dτ P ′ τ − D

satisfies all the conditions. If the dimensions of all the fibers are zero, we could take AP = P − τ P τ − D.


13

0 (B) and at least one component of the fiber has the (i) Assume that ind(D) = 0 ∈ KG nonzero dimension. As the proof of (ii) for the odd case, for r > 0 fixed, we can choose an equivariant spectral section P ′ in the odd sense such that P ′ = 0 on Pλ≤r (D). From (2.2), we have τ P ′ τ = 0 on Pλ≥−r (D). Let N = ker(P ′ + τ P ′ τ ). Then N is a finite dimensional equivariant vector bundle over B. We split the vector bundle by N = N+ ⊕ N− . Then 0 (B). The assumption ind(D) = 0 ∈ K 0 (B) implies that there ind(D) = [N+ ] − [N− ] ∈ KG G exists a vector bundle U such that N+ ⊕ U ≃ N− ⊕ U . We choose another equivariant spectral section P ′′ in the odd sense such that Range(P ′ −P ′′ ) is an equivariant vector bundle whose rank is large enough. Let N ′ = ker(P ′′ + τ P ′′ τ ) and N ′ = N+′ ⊕ N−′ . Let W± be the vector bundles such that N±′ = N± ⊕ W± . Then D+ induces an isomorphism between W+ and W− . Since the rank of W± is large enough, there exist subbundles U+ ⊂ W+ and U− ⊂ W− such that U+ ≃ U− ≃ U. So N+′ ≃ N−′ as vector bundles. 0 (B), the isomorphism is G-equivariant. Let φ : N ′ → N ′ be an Since ind(D) = 0 ∈ KG − + equivariant unitary bundle isomorphism. Using this to write operators on N ′ as 2 × 2 matrices 1 1 i , PN = 2 −i 1

we obtain an equivariant spectral section P = P ′′ + PN . The other direction follows from (iii) easily. The proof of Proposition 2.3 is complete.

Remark 2.4. From the proof of Proposition 2.3 (i) in the even case, we note that for any F ∈ F∗G (B), even for the zero dimensional case, there exists an equivariant spectral section for F + F op . If F is odd and R, P are two equivariant spectral sections with respect to an equivariant B-family D such that P R = R, then coker{Pb Rb : Im(Rb ) → Im(Pb )}b∈B forms an equivariant vector bundle over B, denoted by [P − R]. Hence for any two equivariant spectral sections P , 0 (B) as follows: Q, the difference element [P − Q] can be defined as an element in KG (2.5)

0 [P − Q] := [P − R] − [Q − R] ∈ KG (B),

where R is any equivariant spectral section as in Proposition 2.3 (ii) such that P R = R and QR = R. We note that the class in (2.5) is independent of the choice of R. From the definition in (2.5), we can obtain that if P1 , P2 , P3 are equivariant spectral sections with respect to D, then (2.6)

0 [P3 − P1 ] = [P3 − P2 ] + [P2 − P1 ] ∈ KG (B).

Recall that in Definition 1.2, F ≃ F ′ if they satisfy five conditions. If only the first three conditions hold, we say that F and F ′ have the same topological structure. Note that a horizontal subbundle on W is simply a splitting of the exact sequence (2.7)

0 → T Z → T W → π ∗ T B → 0.

As the space of the splitting map is affine, it follows that any pair of equivariant horizontal subbundles can be connected by a smooth path of equivariant horizontal distributions. Assume that F, F ′ ∈ F∗G (B) have the same topological structure. Let r ∈ [0, 1] parametrize a ′H H TZ H H H W} smooth path {Tπ,r r∈[0,1] such that Tπ,0 W = Tπ W and Tπ,1 W = Tπ W . Similarly, let gr , Z and hE be the G-invariant metrics on T Z, L hL Z and E, depending smoothly on r ∈ [0, 1], r r ′ ′ ′ E T Z L Z which coincide with g , h and h at r = 0 and with g T Z , h LZ and h E at r = 1. By the

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E Z same reason, we can choose G-invariant Hermitian connection ∇L r and ∇r on LZ and E, such ′ ′ L L E E E LZ , ∇ Z = ∇ LZ . Z that ∇E 0 = ∇ , ∇1 = ∇ , ∇0 = ∇ 1 e = [0, 1]×B and pr : B e → B be the projection. We consider the bundle π f := [0, 1]× Let B e:W H H W e f f W → B together with the canonical projection Pr : W → W . Then Tπe W(r,·) = R × Tπ,r f , and T Z e := Pr∗ T Z, L fZ := Pr∗ LZ and defines an equivariant horizontal subbundle of T W e e f e := Pr∗ E are naturally equipped with metrics gT Z , hLZ , hE and connections ∇LfZ , ∇Ee . Then E f→B e such that G acts as identity the fiberwise G-action can be naturally extended to π e:W e e f e f E E L T Z L Z Z on [0, 1] and g , h , h , ∇ , ∇ are G-invariant. Thus, we get two equivariant geometric families

(2.8)

E E H LZ Z Fr = (W, LZ , E, o, Tπ,r W, grT Z , hL r , ∇r , hr , ∇r )

and (2.9)

f, L fZ , E, e o, T H W f , gT Ze , hLfZ , ∇LfZ , hEe , ∇Ee ) Fe = (W π e

such that F0 = F and F1 = F ′ . Now we consider a continuous family of operators Dr on Fr for r ∈ [0, 1] such that for any r ∈ [0, 1], Dr is an equivariant B-family on Fr . Assume that the index of D(F) vanishes. Then the homotopy invariance of index implies that the indice of Dr vanish. Let Q0 , Q1 be equivariant spectral sections with respect to D0 , D1 respectively. If we consider the total family e = {Dr } parametrized by B × I, then there is a total equivariant spectral section Pe = {Pr }. D Let Pr be the restriction of Pe over B × {r}. Thus we have the natural equivariant extension of [16, Definition 1.5]

Definition 2.5. If F, F ′ ∈ F1G (B), the equivariant Dai-Zhang higher spectral flow sf G {(D0 , Q0 ), 0 (B) defined by (D1 , Q1 )} between the pairs (D0 , Q0 ), (D1 , Q1 ) is an element in KG (2.10)

0 sf G {(D0 , Q0 ), (D1 , Q1 )} = [Q0 − P0 ] − [Q1 − P1 ] ∈ KG (B).

From (2.6), we know that this definition is independent of the total equivariant spectral section Pe. In the following, we define the equivariant higher spectral flow for the even case. Let F ∈ F0G (B). Let D be an equivariant B-family on F. We assume that there exists an equivariant spectral section P with respect to D. Let AP be the smooth operator associated with P . We take the odd equivariant geometric family p∗1 F ∪ p∗2 F L as in Example 1.4 d). b in F. Let DL = τ ⊗ D(F L ) on L over p∗1 W ×B×S 1 Let τ be the Z2 -grading of the SZ ⊗E ∗ 1 1 p2 (S × S ) (compare with the notation of Clifford algebra in Section 1.1). Let DP = (D +AP )⊗1+DL . Then DP is an equivariant B ×S 1 -family on the odd geometric family p∗1 F ∪ p∗2 F L and commutes with the group action. Since D and AP aniti-commutes with τ , we have [(D +AP )⊗1, DL ] = 0. So DP2 = ((D +AP )⊗1+DL )2 = (D +AP )2 ⊗1+DL2 > 0. It implies that DP is invertible. Let P ′ be the APS projection for DP . Then P ′ is an equivariant spectral section with respect to the equivariant B × S 1 -family DP . Similarly, Let Q be another equivariant spectral section of D, we can get the equivariant spectral section Q′ with respect to the equivariant B × S 1 -family DQ as above. Since p∗1 F ∪ 0 (B × S 1 ). p∗2 F L ∈ F1G (B), from Definition 2.5, we could get sf G {(DP , P ′ ), (DQ , Q′ )} ∈ KG Now we consider the Example 1.4 c) more explicitly. It is easy to calculate that for θ ∈ [0, 1) fixed, the eigenvalues of D(F L ) are λk (θ) = 2πk + 2π(θ − 1/2), k ∈ Z. So for θ ∈ S 1 , θ 6= 1/2, we have DL2 > 0. Thus for any s ∈ [0, 1], θ 6= 1/2, restricted on B × {θ}, (1 − s)DP + sDQ is invertible. From Definition 2.5, it means that for θ 6= 1/2, sf G {(DP , P ′ ), (DQ , Q′ )}|B×{θ} =


15

0 (B × {θ}). In the sense of (1.12), we have sf {(D , P ′ ), (D , Q′ )} ∈ K 1 (B). We define 0 ∈ KG G P Q G

(2.11)

1 [P − Q] := sf G {(DP , P ′ ), (DQ , Q′ )} ∈ KG (B).

The idea for this construction comes from [29, Proposition 4]. We note that when the group G is trivial, this definition is equivalent to that there. Similarly, if P1 , P2 , P3 are equivariant spectral sections with respect to D, then (2.12)

1 [P3 − P1 ] = [P3 − P2 ] + [P2 − P1 ] ∈ KG (B).

Now we extend the difference [P − Q] to the equivariant higher spectral flow. Let F, F ′ ∈ F0G (B) which have the same topological structure and D0 , D1 be two equivariant B-families on F, F ′ respectively. Let Q0 , Q1 be equivariant spectral sections with respect to D0 , D1 respectively. Let D(r), r ∈ [0, 1] be a continuous curve of equivariant B-families on Fr such e = {D(r) ⊗ 1 + D L } parametrized by that D(0) = D0 + AQ0 and D(1) = D1 + AQ1 . Let D 1 L 1 1 e = 0 ∈ K 1 (B ×S 1 ×I). B ×S ×I. Since ind(D(0)⊗1+D ) = 0 ∈ KG (B ×S ), we have ind(D) G e When restricted Let Pe = {P (r)}r∈[0,1] be an equivariant spectral section with respect to D. e B×{θ}×I is invertible. Let {P ′ (r)θ }r∈[0,1] be the APS projection on B × {θ} × I for θ 6= 1/2, D| e B×{θ}×I . Then P ′ (0)θ = Q′ |B×{θ} and P ′ (1)θ = Q′ |B×{θ} . Since [P ′ (r)θ − P (r)|B×{θ}×I ] of D| 0 1 is an equivariant vector bundle over B × {θ} × I, we have ([Q′0 − P (0)] − [Q′1 − P (1)])|B×{θ} = 0 (B × {θ}). It implies that 0 ∈ KG (2.13)

1 sf G {(D0 ⊗ 1 + D L , Q′0 ), (D1 ⊗ 1 + D L , Q′1 )} = [Q′0 − P (0)] − [Q′1 − P (1)] ∈ KG (B).

Definition 2.6. The equivariant higher spectral flow sf G {(D0 , Q0 ), (D1 , Q1 )} between the pairs 1 (B) defined by (D0 , Q0 ), (D1 , Q1 ) is an element in KG (2.14)

1 sf G {(D0 , Q0 ), (D1 , Q1 )} := sf G {(D0 ⊗ 1 + D L , Q′0 ), (D1 ⊗ 1 + D L , Q′1 )} ∈ KG (B).

Note that when F = F ′ , D0 = D1 , the equivariant higher spectral flow is [Q0 − Q1 ]. From (2.6), we know that this definition is independent of the total equivariant spectral section Pe. Now we will represent the element of equivariant K-group as equivariant higher spectral flow. The proof of the following proposition is constructive. 0 (B) (resp. K 1 (B)), there exist F ∈ F1 (B) (resp. F0 (B)) Proposition 2.7. For any x ∈ KG G G G and equivariant spectral sections P , Q with respect to D(F), such that [P − Q] = x. 0 (B) could be represented as an equivariant virtual bundle E − E . Proof. Any element of KG + − 1 Let π : B × S → B be the projection on the first part. Let F0 = (B × S 1 , π ∗ (E± ), o, T H (B × 1 1 S 1 ), gT S , π ∗ hE± , π ∗ ∇E± ) ∈ F1G (B), where o ∈ o(T S 1 ), gT S are the canonical orientation and metric on S 1 and T H (B × S 1 ) = T B × S 1 . Let ∂t be the generator of T S 1 . Then √ D(F0 ) = − −1∂t · IdE± . We could calculate that the eigenvalues of D(F0 ) are λk = k for k ∈ Z. We denote by Pλ≥k the orthogonal projection onto the union of the eigenspaces of λ ≥ k. Then for any k, Pλ≥k is an equivariant spectral section of D(F0 ). In particular, we 0 (B). have [Pλ≥k − Pλ≥k+1 ] = [E+ ] − [E− ] ∈ KG 1 For any x ∈ KG (B), as in the proof of Proposition 1.5, there exists a finite dimensional G-representation V , such that x can be represented as a G-invariant unitary element F ∈ End(V ) ⊗ C 0 (B, C). Let F1 = (B, E+ = E− = B × V, hE± , ∇E± ) ∈ FG0 (B), where hE± and ∇E± are trivial. Set 0 I 0 F (b)∗ , A1 = . A0 = F (b) 0 I 0

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Let Pi be the orthogonal projection onto the positive part of the spectrum of Ai for i = 0, 1. From Definition 2.2, we know that P0 and P1 are equivariant spectral sections with respect to D(F1 ) = 0 on F1 . Let Di = Ai ⊗ 1 + τ ⊗ D(F L ) on p∗1 F1 ∪ p∗2 F L and Pi′ be the APS projection of Di . Let Ds = (1 − s)D0 + sD1 for s ∈ [0, 1]. We claim that 0 sf G {(D0 , P0′ ), (D1 , P1′ )} = [W ] − [U ] ∈ KG (B × S 1 ),

(2.15)

where W and U are bundles constructed in the proof of Proposition 1.5. Then from the proof of Proposition 1.5, (2.11) and (2.15), we obtain Proposition 2.7 in the odd case. We prove the claim (2.15) constructively. Let λb,i be the eigenvalues of F (b) on V with ± eigenvectors vb,i . Then λb,i is the eigenvalues of F ∗ (b) on V with eigenvectors vb,i . Let vb,i be the corresponding vector q in E± . Let sk (θ) be the eigenvector of λk (θ) with respect to D(F L ). Let λs,b,i,k (θ) = λk (θ)2 + s2 + (1 − s)2 + s(1 − s)(λb,i + λb,i ). It is easy to calculate that {λs,b,i,k } are the set of nonnegative eigenvalues of Ds . Let (1)

− + + (λs,b,i,1 (θ) − λ1 (θ))−1 ((1 − s)λb,i + s)vb,i ) ⊗ s1 (θ), us,b,i (θ) = (vb,i

for 0 ≤ θ ≤ 1/2,

(2)

+ − us,b,i (θ) = (vb,i + (λs,b,i,−1 (θ) + λ−1 (θ))−1 ((1 − s)λb,i + s)vb,i ) ⊗ s−1 (θ),

(2.16)

for 1/2 ≤ θ ≤ 1,  − + −1  (v + (λs,b,i,0 (θ) − λ0 (θ)) · ((1 − s)λb,i + s)vb,i ) ⊗ s0 (θ),   b,i (3) for 0 ≤ θ ≤ 1/2, λs,b,i,0 (θ) 6= 0, us,b,i (θ) =    v − ⊗ s (θ), for 0 ≤ θ ≤ 1/2, λs,b,i,0 (θ) = 0, 0 b,i  − + −1   (vb,i + (λs,b,i,0 (θ) + λ0 (θ)) · ((1 − s)λb,i + s)vb,i ) ⊗ s0 (θ),  (4) for 1/2 ≤ θ ≤ 1, λs,b,i,0 (θ) 6= 0. us,b,i (θ) =    v + ⊗ s (θ), for 1/2 ≤ θ ≤ 1, λ (θ) = 0 . 0

b,i

s,b,i,0

(2)

(1)

Then we can calculate that us,b,i and us,b,i are eigenfunctions of λs,b,i,1 (θ) with respect to Ds (3)

(4)

and us,b,i and us,b,i are that of λs,b,i,0 (θ). Choose χ(θ) ∈ C ∞ ([0, 1/2]) with χ(θ) = 1/2 at θ = 0 and χ(θ) = 0 at θ = 1/2. Let (3)

(1)

(5)

us,b,i (θ) = χ(θ)us,b,i + (1 − χ(θ))us,b,i ,

(2.17)

(2)

(6)

for 0 ≤ θ ≤ 1/2, (4)

us,b,i (θ) = χ(1 − θ)us,b,i + (1 − χ(1 − θ))us,b,i , (5)

(6)

(5)

for 1/2 ≤ θ ≤ 1.

(6)

We have us,b,i (0) = us,b,i (1). So span{us,b,i } and span{us,b,i } make a trivial equivariant vector (5)

(6)

bundle over B × (S 1 \{1/2}) × [0, 1]. Thus the relations us,b,i (1/2) ∼ us,b,i (1/2) makes an f over B × S 1 × [0, 1]. Let R e be the orthogonal projection onto the equivariant vector bundle W f and the eigenspaces with non-positive eigenvalues of D e = {Ds }. Then Q e = 1−R e is sum of W ′ e s=0 ] = W and e It is easy to see that [P − Q| an equivariant spectral section with respect to D. 0 ′ e [P − Q|s=1 ] = U . So we obtain the claim (2.15). 1

The proof of Proposition 2.7 is complete.

Note that the proof of Proposition 2.7 in odd case gives a nontrivial example of equivariant higher spectral flow for even dimensional fibers and an example of equivariant spectral section without the spectral gap.


17

Remark 2.8. If G = {e}, there is a stronger version of Proposition 2.7 (see [29, Proposition 12]): for any x ∈ K 0 (B) (resp. K 1 (B)), F ∈ F1 (B) (resp. F0 (B)) and spectral section P with respect to D(F), there exists spectral section Q with respect to D(F), such that [P − Q] = x. 2.2. Equivariant local family index theorem. In this subsection, we use the notations in Section 1.2 and describe the equivariant local index theorem for F ∈ F∗G (B) when the G-action on B is trivial. b b . As in [5], we will regard For b ∈ B, let Eb be the set of smooth sections over Zb of SZ ⊗E E as an infinite dimensional fiber bundle over B. Let 0 ∇T W be the connection on T W = T H W ⊕ T Z defined by 0

(2.18)

∇T W = π ∗ ∇T B ⊕ ∇T Z .

Then 0 ∇T W preserves the metric gT W in (1.6). Set

S = ∇T W − 0 ∇T W .

(2.19)

If V ∈ T B, let V H ∈ TπH W be its horizontal lift in TπH W so that π∗ V H = V . For any b V ∈ T B, s ∈ C ∞ (B, E ) = C ∞ (W, SZ ⊗E), by [7, Proposition 1.4], the connection

1 b ∇EV ,U s := ∇SVZH⊗E s − hS(ei )ei , V H i s 2 2 preserves the L -product on E . Let {fp } be a local orthonormal frame of T B and {f p } be its dual. We denote by ∇E ,U = p f ∧ ∇Efp,U . Let T be the torsion of 0 ∇T W . Then T (fpH , fqH ) ∈ T Z. We denote by (2.20)

1 c T (fpH , fqH ) f p ∧ f q ∧ . 2 By [5, (3.18)], the (rescaled) Bismut superconnection c(T ) =

(2.21)

b ) → C ∞ (B, Λ(T ∗ B)⊗E b ) Bu : C ∞ (B, Λ(T ∗ B)⊗E

(2.22) is defined by

Bu =

(2.23)

√ 1 uD(F) + ∇E ,U − √ c(T ). 4 u

Obviously, the Bismut superconnection Bu commutes with the G-action. Moreover, B2u is a 2-order elliptic differential operator along the fibers Z. Let exp(−B2u ) be the family of heat operators associated with the fiberwise elliptic operator B2u . From [4, Theorem 9.50], we know that exp(−B2u ) is a smooth family of smoothing operators. b End(EZ ) which takes value in Λ(T ∗ B), we If P is a trace class operator acting on Λ(T ∗ B)⊗ use the convention that if ω ∈ Λ(T ∗ B),

(2.24)

Trs [ωP ] = ω Trs [P ].

odd/even

We denote by Trs (2.25)

[P ] the part of Trs [P ] which takes value in odd or even forms. Set if dim Z is even; f ] = Trs [P ], Tr[P Trodd [P ], if dim Z is odd. s

We assume that G acts trivially on B. Take g ∈ G. Let W g be the fixed point set of g on W . Set √ −1 E E (2.26) R |W g . chg (E, ∇ ) = Trs g exp 2π

18

BO LIU

Let chg (E) ∈ H even (W g , C) denote the cohomology class of chg (E, ∇E ). When Z = pt, it descends to the equivariant Chern character map 0 chg : KG (B) −→ H even (B, C).

(2.27)

1 (B), we can regard x as an element x′ in K 0 (B×S 1 ). The odd equivariant By (1.12), for x ∈ KG G Chern character map 1 chg : KG (B) −→ H odd (B, C)

(2.28) is defined by (2.29)

chg (x) =

Z

S1

chg (x′ ).

We adopt the sign notation in the integral as in (0.8). Set √ 1/2 −1 LZ 1/2 LZ chg (LZ , ∇ ) = Trs g exp R |W g (2.30) . 4π Let ∇ be a Euclidean connection on (T Z, gT Z ). We denote by 1/2

b g (T Z, ∇) ∧ chg (L1/2 , ∇LZ ). Tdg (∇, ∇LZ ) := A Z

(2.31)

b g , see [23, (1.44)]. Let Tdg (T Z, LZ ) ∈ H even (W g , C) For the definitions of the characteristic form A L Z denote the cohomology class of Tdg (∇, ∇ ). For α ∈ Ωi (B), set  i 2  1  √ · α, if i is even; 2π −1 (2.32) ψB (α) = i−1  2 1  √1 √ · α, if i is odd. π 2π −1 We state the equivariant family local index theorem [23, Theorem 1.2] here. Note that π : W g → B is a fiber bundle with closed fiber Z g . From [23, Proposition 1.1], Z g is naturally oriented.

f exp(−B2u )] ∈ Ω∗ (B, C) Theorem 2.9. For any u > 0 and g ∈ G, the differential form ψB Tr[g ∗ is closed and its cohomology class represents chg (ind(D(F))) ∈ H (B, C). As u → 0, we have Z 2 f (2.33) Tdg (∇T Z , ∇LZ ) ∧ chg (E, ∇E ). lim ψB Tr[g exp(−Bu )] = u→0

Zg

To simplify the notations, we set

(2.34)

FLIg (F) =

Z

Zg

Tdg (∇T Z , ∇LZ ) ∧ chg (E, ∇E ).

0/1

So Theorem 2.9 says that for F ∈ FG (B) [FLIg (F)] = chg (ind(D(F))) ∈ H even/odd (B, C).

(2.35)

When F is the equivariant geometric family in Example 1.4 a), the equivariant family local index theorem degenerates to the equivariant Chern-Weil theory: (2.36)

f exp(−B2u )] = ψB Trs [g exp(−∇E,2 )] = chg (E, ∇E ). lim ψB Tr[g

u→0

In this case, FLIg (F) = chg (E, ∇E ) = chg (E+ , ∇E+ ) − chg (E− , ∇E− ).


19

2.3. Equivariant eta form. In this subsection, we also assume that G acts trivially on B. We define the equivariant Bismut-Cheeger eta form with perturbation operator. Definition 2.10. Let D be an equivariant B-family on F. A perturbation operator with respect to D is a family of bounded pseudodifferential operators A such that D + A is an invertible equivariant B-family on F. Note that if there exists an equivariant spectral section with respect to D, the smooth operator associated with it is a perturbation operator. In this subsection, we assume that there exists a perturbation operator with respect to D(F) on F. It will imply that ind(D(F)) = 0. Let χ ∈ C ∞ (R) be a cut-off function such that ( 0, if u < 1; χ(u) = (2.37) 1, if u > 2. Let A be a perturbation operator with respect to D(F). For g ∈ G, set √ √ (2.38) B′u = Bu + uχ( u)A. √ Since χ( u) = 0 when u → 0, by (2.34), f exp(−(B′u )2 )] = FLIg (F) ∈ Ω∗ (B, C). lim ψB Tr[g

(2.39)

u→0

√ Since χ( u) = 1 when u → +∞, from [4, Theorem 9.19], we have f g exp −(B′u )2 = 0. lim ψB Tr (2.40) u→+∞

If α ∈

(2.41)

Λ(T ∗ (R

+

× B)), we can expand α in the form α = du ∧ α0 + α1 ,

α0 , α1 ∈ Λ(T ∗ B).

Set (2.42)

[α]du = α0 .

Definition 2.11. For any g ∈ G, modulo exact forms, the equivariant eta form with perturbation operator A is defined by " 2 !#)du Z ∞( ∂ f g exp − B′ + du ∧ ψB Tr (2.43) η˜g (F, A) = − du u ∂u 0 ∈ Ω∗ (B, C)/dΩ∗ (B, C).

The regularities of the integral in the right hand side of (2.43) are proved in [23, Section 1.4]. As in [23, (1.81)], we have (2.44)

d˜ ηg (F, A) = FLIg (F).

From the proof of the anomaly formula in [23, Theorem 1.7], it is easy to see that the value of ηeg (F, A) in Ω∗ (B, C)/dΩ∗ (B, C) is independent of the choice of the cut-off function. Similarly, If AP and A′P are two smooth operators associated with the same equivariant spectral section P , we have ηeg (F, AP ) = ηeg (F, A′P ) ∈ Ω∗ (B, C)/dΩ∗ (B, C). In this case, we often simply denote it by ηeg (F, P ). Note that the taming in [10, 12, 14] is a perturbation operator. So the definition here is also a generalization of eta forms there.

20

BO LIU

If the fiber Z is connected, we could calculate the equivariant eta form explicitly:

Z ∞ ∂B′u 1  ′ 2 even  √  exp(−(B ) ) du ∈ Ω∗ (B, C)/dΩ∗ (B, C), ψ Tr g B u s   ∂u π  0     if F is odd;    Z  ∞  1 ∂B′u   √ √ ψB Trs g exp(−(B′u )2 ) du ∈ Ω∗ (B, C)/dΩ∗ (B, C), ∂u 2 −1 π (2.45) η˜g (F, A) = 0    if F is even and dim Z > 0.    √ Z  ∞ 2 E E  (∇ ) −1 ∂∇u    Trs g exp − √u du ∈ Ω∗ (B, C)/dΩ∗ (B, C),   0 2π ∂u 2π −1     if dim Z = 0, √ √ E where ∇E u = ∇ + uχ( u)A. When dim Z = 0, the equivariant geometric family degenerates to the case of Example 1.4 a). As in [27, Definition B.5.3], from (2.36) and (2.44), the equivariant eta form in this case is just the equivariant transgression between chg (E, ∇E ) and 0. Furthermore, by changing the variable (see also [23, Remark 1.4]), we could get another form of equivariant eta form:

(2.46)

η˜g (F, A) = −

Z

0

∞

(

" !#)du ∂ 2 ′ f ψB Tr g exp − Bu2 + du ∧ du. ∂u ′

′

Let (Z ′ , g T Z ) be a Spinc manifold with even dimension and (E ′ , hE ) be a Z2 -graded Her′ mitian vector bundle over Z ′ with Hermitian connection ∇E . Let pr2 : B × Z ′ → Z ′ be the projection onto the second part. Then all the bundles and geometric data above could be pulled back on B × Z ′ . Thus the fiber bundle pr1 : B × Z ′ → B, which is the projection onto the first part, and the structures pulled back by pr2 form a geometric family F ′ with fibers Z ′ . We assume that the group action on F ′ is trivial. In this case, ind(D(F ′ )) is a constant integer. For F ∈ F∗G (B), let A be a perturbation operator with respect to D(F) on F. Then b is a perturbation operator with respect to D(F ∪ F ′ ) on F ∪ F ′ . A⊗1 Lemma 2.12. For g ∈ G, we have

(2.47)

b = η˜g (F, A) · ind(D(F ′ )). η˜g (F ∪ F ′ , A⊗1)


21

Proof. We denote by Tr |F the trace operator on F. Then b (2.48) η˜g (F ∪ F ′ , A⊗1) " 2 !#)du Z ∞( ∂ ′ ′ f F ∪F ′ g exp − B 2 ⊗1 b + 1⊗uD(F b ψB Tr| ) + du ∧ du =− u ∂u 0 Z ∞n h io ′ 2 ′ f F ∪F ′ g(1⊗D(F b 2 − 1⊗uD(F b b du ψB Tr| ) )) exp − B′u2 ⊗1 = 0

(

"

!#)du 2 ∂ f F ∪F ′ g exp − B′ 2 ⊗1 b + du ∧ b 2 D(F ′ )2 ψB Tr| − du − 1⊗u u ∂u 0 Z ∞n h o i f F g exp − B′ 2 2 · Trs |F ′ D(F ′ ) exp −u2 D(F ′ )2 du ψB Tr| = u Z

∞

0

−

Z

0

∞

(

fF ψB Tr|

"

)du !# ∂ 2 g exp − Bu2 + du ∧ · Trs |F ′ exp −u2 D(F ′ )2 du ∂u

From the definition of F ′ , we have Trs |F ′ D(F ′ ) exp − u2 D(F ′ )2 = 0, (2.49) Trs |F ′ exp −u2 D(F ′ )2 = ind(D(F ′ )).

b = η˜g (F, A)·ind(D(F ′ )). The proof of Lemma 2.12 is complete. So we get η˜g (F ∪F ′ , A⊗1)

2.4. Anomaly formula. In this subsection, we will study the anomaly formula of the equivariant eta forms for two odd equivariant geometric families F and F ′ with the same topological structure. In this subsection, we also assume that G acts on B trivially. Lemma 2.13. (compare with [28, Proposition 17]) Assume that F ∈ F1G (B). Let A be a family of bounded pseudodifferential operator on F such that D(F) + A is an equivariant B-family. Let P , Q be two equivariant spectral sections with respect to D(F) + A. Let AP , AQ be smooth operators associated with P , Q. For any g ∈ G, modulo exact forms, we have (2.50)

η˜g (F, A + AP ) − η˜g (F, A + AQ ) = chg ([P − Q]) ∈ H even (B, C).

Proof. The proof is the natural equivariant extension of that of [28, Proposition 17]. We sketch it here for the completion. From (2.6), we only need to prove the lemma when Q majorizes P . Let Fe be the equivariant e u be the Bismut geometric family defined in (2.9) such that Fr = F for any r ∈ [0, 1]. Let B superconnection associated with Fe. Choose the smooth operators AP , AQ as in (2.4). Set e u |(u,r) + √uχ(√u)Ar . Let Π be the orthogonal e ′ |(u,r) = B Ar = A + AP + r(AQ − AP ) and B u e and projection onto [P − Q]. Then Range(Π) = [P − Q] is an equivariant vector bundle over B ∇Π := Π ◦ ∇E , U ◦ Π is a G-invariant connection on Range(Π). From (2.3), (2.23) and (2.38), we could calculate that Π ◦ (D(F) + A + Ar ) ◦ Π = s(1 − 2r)Π.

(2.51) So when u large, (2.52)

2 e ′ 22 ◦ Π = u2 s2 (1 − 2r)2 − 2usdr ∧ +Π ∇E , U Π + O(u−1 ). Π◦B u

22

BO LIU

As in the proof of [4, Theorem 9.19] (or following the process in [23, Section 4]), with respect h ′ idr e 22 to the splitting Range(Π) ⊕ Range(Π)⊥ , when u → +∞, we can write exp −B as u

(2.53) Set

h

′ idr e 22 exp −B = u

2use−u

2 s2 (1−2r)2

! 2 + O(u−1 ) O(u−1 ) exp − ∇Π O(u−1 )

O(u−2 )

′ iodr h n odd e 22 g exp −B r1 (u, r) = ψB Trs u

.

.

(u,r)

Then from [23, (1.95)], we have

(2.54) η˜g (F, A + AP ) − η˜g (F, A + AQ ) = − lim 1 = √ lim π u→+∞

Z

u→+∞ 0

Z

1

2use−u

0

2 s2 (1−2r)2

1

r1 (u, r)dr

dr · ψB Tr[g exp(−(∇Π )2 )] = chg ([P − Q]).

The uniformly convergence condition needed in [23, (1.95)] relies on (2.53). The proof of Lemma 2.13 is complete.

From [27, Theorem B.5.4], modulo exact forms, the Chern-Simons forms Z 1 ′T Z ′L f e TZ LZ Z f [Tdg (∇T Z , ∇LZ )]ds ds, Tdg (∇ , ∇ , ∇ , ∇ ) := − 0 (2.55) Z 1 e g (∇E , ∇′ E ) := − e ∇Ee )]ds ds, [chg (E, ch 0

do not depend on the choices of the objects with e. Moreover, ′

(2.56)

′

′

′

f g (∇T Z , ∇LZ , ∇ T Z , ∇ LZ ) = Tdg (∇ T Z , ∇ LZ ) − Tdg (∇T Z , ∇LZ ), dTd ′

′

e g (∇E , ∇ E ) = chg (E, ∇ E ) − chg (E, ∇E ). dch

Let F, F ′ ∈ F∗G (B) which have the same topological structure. By (2.35), we have [FLIg (F)] = [FLIg (F ′ )] ∈ H ∗ (B, C). Set Z ′ f g (∇T Z , ∇LZ , ∇′ T Z , ∇′ LZ ) ∧ chg (E, ∇E ) g Td (2.57) FLIg (F, F ) = Zg Z ′ ′ e g (∇E , ∇′ E ) ∈ Ω∗ (B, C)/dΩ∗ (B, C). Tdg (∇ T Z , ∇ LZ ) ∧ ch + Zg

From (2.56), we have

(2.58)

g g (F, F ′ ) = FLIg (F ′ ) − FLIg (F). d FLI

Using Lemma 2.13, we can get the anomaly formula for odd case as follows. The even case will be proved later. Proposition 2.14. Let F, F ′ ∈ F1G (B) which have the same topological structure. Let A, A′ be perturbation operators with respect to D(F), D(F ′ ) and P , P ′ be the APS projections with respect to D(F) + A, D(F ′ ) + A′ respectively. For any g ∈ G, modulo exact forms, we have g g (F, F ′ ) + chg sf G {(D(F ′ ) + A′ , P ′ ), (D(F) + A, P )} . (2.59) η˜g (F ′ , A′ ) − η˜g (F, A) = FLI


23

Proof. Let Fe be the equivariant geometric family defined in (2.9). Let Dr = D(Fr ) + (1 − e = {Dr }r∈[0,1] on F. e Since the equivariant index of F vanishes, the homotopy r)A + rA′ and D invariance of the equivariant index bundle implies that the equivariant indices of each Dr and e vanish. If we consider the total family Fe, then from Proposition 2.3(i), there exists a total D e Let Pr be the restriction of Pe over {r} × B. equivariant spectral section Pe with respect to D. Let APr be a smooth operator associated with Pr . Following the proof of [23, Theorem 1.7], we can get g g (F, F ′ ). η˜g (F ′ , A′ + AP1 ) − η˜g (F, A + AP0 ) = FLI

(2.60)

Then Proposition 2.14 follows from Lemma 2.13, (2.10) and (2.60).

2.5. Functoriality of equivariant eta forms. In this subsection, we will study the functoriality of the equivariant eta forms and use it to prove the anomaly formula of equivariant eta forms for even equivariant geometric families. In this subsection, we use the notations in Section 1.4. ′ Let ∇ and ∇′ be Euclidean connections on (T Z, g T Z ) and ∇LZ and ∇ LZ be Hermitian connections on (LZ , hLZ ). Similarly as (2.34), we define Z LZ (2.61) Tdg (∇, ∇LZ ) ∧ chg (E, ∇E ). FLIg (∇, ∇ ) := Zg

f g (∇, ∇LZ , ∇′ , ∇′ LZ ) ∈ As in (2.55) and (2.56), there exists a well-defined equivariant Chern-Simons form Td Ω∗ (B, C)/dΩ∗ (B, C) such that

(2.62) Set (2.63)

f g (∇, ∇LZ , ∇′ , ∇′ LZ ) = Tdg (∇′ , ∇′ LZ ) − Tdg (∇, ∇LZ ). d Td

g g (∇, ∇LZ , ∇′ , ∇′ LZ ) := FLI

From (2.62), we have (2.64)

Z

Zg

f g (∇, ∇LZ , ∇′ , ∇′ LZ ) ∧ chg (E, ∇E ). Td

′

′

g g (∇, ∇LZ , ∇′ , ∇ LZ ) = FLIg (∇′ , ∇ LZ ) − FLIg (∇, ∇LZ ). d FLI ′

′

∗ (V ), then ind(D(F )) = 0 ∈ K ∗ +∗ (B). Therefore, Lemma 2.15. If ind(D(FX )) = 0 ∈ KG Z G we get the well-defined property of the push-forward map in Theorem 1.8.

Proof. Let 1 TX g . T2 We denote the Clifford algebra bundle of T Z with respect to gTT Z by CT (T Z). If U ∈ T V , let U H ∈ TπHX W be the horizontal lift of U , so that πX,∗ (U H ) = U . Let {ei }, {fp } be local H } ∪ {T e } is a local orthonormal frame orthonormal frames of (T X, g T X ), (T Y, gT Y ). Then {fp,1 i T Z of (T Z, gT ). We define a Clifford algebra isomorphism (2.65)

(2.66)

∗ TY gTT Z = πX g ⊕

GT : CT (T Z) → C(T Z)

by (2.67)

H H GT (c(fp,1 )) = c(fp,1 ),

GT (cT (T ei )) = c(ei ).

b X is defined in Section 1.1. Under this isomorphism, we can consider Recall that π1∗ SY ⊗S b X ∗ b X )⊗E, b hπ1∗ SY ⊗S ⊗ hE ) as a self-adjoint Hermitian equivariant Clifford module of ((π1 SY ⊗S CT (T Z). Let ∇TT Z be the connection associated with (TπHZ W, gTT Z ) as in (1.7).

24

BO LIU

Then (2.68)

FZ,T = (W, LZ , E, oY ∪ oX , TπHZ W, gTT Z , , hLZ , ∇LZ , hE , ∇E )

is an equivariant geometric family over B and FZ = FZ,1 . ∗ (V ) and at least one component of X has nonzero dimension, from If ind(D(FX )) = 0 ∈ KG Proposition 2.3 (i), there exists a perturbation operator AX such that ker(D(FX ) + AX ) = 0. b X )⊗E) b the same We extend AX to a pseudodifferential operator acting on C ∞ (W, (π1∗ SY ⊗S b way as the extension of c(ei ) in Section 1.1. We denote it by 1⊗AX . As in [23, Lemma 4.3], b AX ) = 0. So by the there exists T ′ ≥ 1, such that when T ≥ T ′ , we have ker(D(FZ,T ) + 1⊗T homotopy invariance of the equivariant index, for any T ≥ 1, we have ind(D(FZ,T )) = 0. If dim X = 0, it is obvious. The proof of Lemma 2.15 is complete. b X is not a smooth operator along the Note that when AX is smooth along the fibers X, 1⊗A fibers Z. This is the reason for us to define the eta form for bounded perturbation operator instead of smooth operator in [10, 12, 16, 28]. Recall that in (1.22), T Z = TπHX Z ⊕ T X. Let ∇T Y,T X be the connection on T Z defined by ∗ ∇T Y,T X = πX ∇T Y ⊕ ∇T X

(2.69)

as in (2.18). The following technical lemma is a modification of the main result in [23]. The proof of it will be left to the next subsection. Lemma 2.16. Let T ′ ≥ 1 be the constant taking in the proof of Lemma 2.15. Let AX be a perturbation operator with respect to D(FX ). Then modulo exact forms, for T ≥ T ′ , we have (2.70) b AX ) = ηeg (FZ,T , 1⊗T

Z

Yg

g g ∇TT Z , ∇LZ , ∇T Y,T X , ∇LZ . Tdg (∇T Y , ∇LY ) ∧ ηeg (FX , AX ) − FLI

Using Lemma 2.16, we could extend the anomaly formula Proposition 2.14 to the general case. Theorem 2.17. Let F, F ′ ∈ F∗G (B) which have the same topological structure. Let A, A′ be perturbation operators with respect to D(F), D(F ′ ) and P , P ′ be the APS projections with respect to D(F) + A, D(F ′ ) + A′ respectively. For any g ∈ G, modulo exact forms, we have (2.71)

g g (F, F ′ ) + chg sf G {(D(F ′ ) + A′ , P ′ ), (D(F) + A, P )} . η˜g (F ′ , A′ ) − η˜g (F, A) = FLI

Proof. We only need to prove the even case. We will add an additional dimension as in Example 1.4 d) such that the new family is odd. Let L → S 1 × S 1 be the Hermitian line bundle in Example 1.4 c) with ∇L constructed there. Let pr2 : B × S 1 × S 1 → S 1 × S 1 be the natural projection. Then all the bundles and geometric data could be pulled back on B × S 1 × S 1 . Thus the fiber bundle pr1 : B × S 1 × S 1 → B, which is the projection onto the first part, and the structures pulled back by pr2 make a geometric family F0 . In this case, ind(D(F0 )) = 1. The key observation is (2.72)

p1 !(p∗1 F ∪ p∗2 F L ) = F ∪ F0 .


25

b F0 is a perturbation operator of D(F ∪ F0 ), we could choose T ′ = 1 in Lemma Since A⊗1 2.16. So by Lemma 2.12 and 2.16, we have b F0 ) (2.73) η˜g (F, A) = η˜g (F ∪ F0 , A⊗1 Z g g ∇T (Z×S 1 ) , ∇LZ , ∇T Z,T S 1 , ∇LZ . b F L ) − FLI η˜g (p∗1 F ∪ p∗2 F L , A⊗1 = S1

Note that D(p∗1 F ∪ p∗2 F L ) = D(F) ⊗ 1 + D L . By Proposition 2.14, the construction of the equivariant higher spectral flow for even case and (2.29), we have Z b FL ) b F L ) − η˜g (p∗1 F ∪ p∗2 F L , A⊗1 (2.74) η˜g (F ′ , A′ ) − η˜g (F, A) = η˜g (p∗1 F ′ ∪ p∗2 F L , A′ ⊗1 S1 Z Z n o f g ∇T (S 1 ×Z) , ∇LZ , ∇T S 1 ,T Z , ∇LZ f g ∇T (S 1 ×Z ′ ) , ∇LZ ′ , ∇T S 1 ,T Z ′ , ∇LZ ′ − Td Td + S1 Zg Z Z f g ∇T S 1 ,T Z , ∇LZ , ∇T S 1 ,T Z ′ , ∇LZ ′ Td = S1 Zg Z b P0 )} b P0′ ), (D(p∗1 F ∪ p∗2 F L ) + A⊗1, chg sf G {(D(p∗1 F ′ ∪ p∗2 F L ) + A′ ⊗1, + 1 ZS f g ∇T Z , ∇LZ , ∇T Z ′ , ∇LZ ′ + chg sf G {(D(F ′ ) + A′ , P ′ ), (D(F) + A, P )} , Td = Zg

where P0 , P0′ are the associated APS projections respectively. Note that in order to adapt the sign convention (0.8), the sign in the beginning of the second line of (2.74) is alternated. The proof of Theorem 2.17 is complete. Using Theorem 2.17, we could write the Lemma 2.16 as a more elegant form. Theorem 2.18. Let AZ and AX be perturbation operators with respect to D(FZ ) and D(FX ). Then modulo exact forms, for T ≥ 1 large enough, we have Z g g ∇T Z , ∇LZ , ∇T Y,T X , ∇LZ Tdg (∇T Y , ∇LY ) ∧ ηeg (FX , AX ) − FLI (2.75) ηeg (FZ , AZ ) = Yg

b AX , P ′ )}), + chg (sf G {(D(FZ ) + AZ , P ), (D(FZ,T ) + 1⊗T

where P and P ′ are the associated APS projections respectively.

From Theorem 2.17 and 2.18, we could extend Lemma 2.12 to the general case. Theorem 2.19. (compare with [12, (24)]) Let F, F ′ ∈ F∗G (B). Let A and A′ be the perturbation ∗ (B), such that operators with respect to D(F) and D(F ∪ F ′ ). Then there exists x ∈ KG ηeg (F ∪ F ′ , A′ ) = ηeg (F, A) ∧ FLIg (F ′ ) + chg (x).

(2.76)

Proof. Here we use a trick in [12] similarly as (2.72). Let π ′ : W ′ → B be the submersion in ′ ′ F ′ . We could get the pullback π ∗ F. Let π ∗ F ⊗ E ′ be the equivariant geometric family which ′ is obtained from π ∗ F by twisting with δ∗ (SZ ′ ⊗ E ′ ), where δ : W ×B W ′ → W ′ . Then we have ′

F ∪ F ′ ≃ π ′ !(π ∗ F ⊗ E ′ ).

(2.77) ′

′

′

Since the fibers of π ∗ W → B is Z ′ × Z, the fiberwised connection ∇T (Z ×Z) = ∇T Z ,T Z . So Theorem 2.19 follows from Theorem 2.18. The proof of Theorem 2.19 is complete.

26

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Remark 2.20. When the parameter space B is a point, dim Z is odd, letting A = Pker D be the orthogonal projection onto the kernel of D(F), which we simply denote by D Z +∞ 1 (2.78) η˜g (F, A) = √ Tr g(D + (uχ(u))′ Pker D ) exp(−(uD + uχ(u)Pker D )2 ) du π 0 Z +∞ 1 =√ Tr g(D + (uχ(u))′ Pker D ) exp(−u2 D − u2 χ(u)2 Pker D ) du π 0 Z +∞ Z +∞ 1 1 2 2 =√ Tr gD exp(−u D ) du + √ Tr g(uχ(u))′ Pker D exp(−u2 χ(u)2 Pker D ) du π 0 π 0 Z +∞ Z +∞ 1 1 −1/2 2 = √ u Tr[gD exp(−uD )]du + √ exp(−u2 )du · Tr[gPker D ] 2 π 0 π 0 Z +∞ 1 1 √ u−1/2 Tr[gD exp(−uD 2 )]du + Tr[gPker D ], = 2 π 0 2 which is just the usual equivariant reduced eta invariant [18]. So Theorem 2.17 and 2.18 naturally degenerate to the case of equivariant reduced eta invariants and the higher spectral flow degenerates to the canonical spectral flow. Even for this case, our results are not found in the literatures. 2.6. Proof of Lemma 2.16. The proof of Lemma 2.16 is almost the same as the proof of [23, Theorem 2.4] and Assumption 2.1 and 2.3 in [23] naturally hold in our case. Let T ′ ≥ 1 be the constant taking in the proof of Lemma 2.15. For T ≥ T ′ , let Bu,T be the Bismut superconnection associated with equivariant geometric family FZ,T . Let (2.79)

∂ ∂ b (T,u) = Bu2 ,T + uT χ(uT )(1⊗A b X ) + dT ∧ + du ∧ . B| ∂T ∂u

b 2 )] of degree one with respect f exp(−B We define βg = du∧ βgu + dT ∧ βgT to be the part of ψB Tr[g to the coordinates (T, u), with functions βgu , βgT : R+,T × R+,u → Ω∗ (B, C). Comparing with [23, Proposition 3.2], there exists a smooth family αg : R+,T × R+,u → ∗ Ω (B, C) such that ∂ ∂ du ∧ + dT ∧ (2.80) βg = dT ∧ du ∧ dS αg . ∂u ∂T Take ε, A, T0 , 0 < ε ≤ 1 ≤ A < ∞, T ′ ≤ T0 < ∞. Let Γ = Γε,A,T0 be the oriented contour in R+,T × R+,u . u A Γ3

ε 0

Γ2

Γ

U

Γ1

Γ4

T′

T0

T


27

The contour Γ is made of four oriented pieces Γ1 , · · · , Γ4 indicated in the above picture. For R 0 1 ≤ k ≤ 4, set Ik = Γk βg . Then by Stocks’ formula and (2.80), (2.81)

4 X k=1

Ik0

=

Z

βg = ∂U

Z U

∂ ∂ du ∧ + dT ∧ ∂u ∂T

S

βg = d

Z

U

αg dT ∧ du .

The following theorems are the analogues of Theorem 3.3-3.6 in [23]. We will sketch the proofs in the next subsection. Theorem 2.21. i) For any u > 0, we have lim βgu (T, u) = 0.

(2.82)

T →∞

ii) For 0 < u1 < u2 fixed, there exists C > 0 such that, for u ∈ [u1 , u2 ], T ≥ 1, we have |βgu (T, u)| ≤ C.

(2.83) iii) We have the following identity: (2.84)

lim

Z

T →+∞ 1

∞

βgu (T, u)du = 0.

Theorem 2.22. We have the following identity: Z ∞ (2.85) βgT (T, u)dT = 0. lim u→+∞ 1

b g (T Z, ∇) only depends on g ∈ G and R := ∇2 . So we can denote it by We know that A T Z b g (R). Let R := (∇T Z )2 . Set A T T TZ ∂∇ ∂ T Z T bg R + b γΩ (T ) = − A (2.86) . T ∂b b=0 ∂T By a standard argument in Chern-Weil theory, we know that (2.87)

∂ e TZ b g (T Z, ∇T Z A T ′ , ∇T ) = −γΩ (T ). ∂T

Proposition 2.23. When T → +∞, we have γΩ (T ) = O(T −2 ). Moreover, modulo exact forms on W g , we have Z +∞ e TZ T Y,T X b (2.88) γΩ (T )dT. Ag (T Z, ∇T ′ , ∇ )=− T′

Let BX,T be the Bismut superconnection associated with the equivariant geometric family FX . Set ( " 2 !#)dT ∂ f V g g exp − BX,T 2 |V g + T χ(T )AX |V g + dT ∧ (2.89) . γ1 (T ) = ψV g Tr| ∂T Then

(2.90)

ηeg (FX , AX ) = −

Z

∞

γ1 (T )dT.

0

Theorem 2.24. i) For any u > 0, there exist C > 0 and δ > 0 such that, for T ≥ T ′ , we have (2.91)

|βgT (T, u)| ≤

C

T 1+δ

.

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ii) For any T > 0, we have (2.92)

lim

ε→0

ε−1 βgT (T ε−1 , ε)

=

Z

Yg

Tdg (∇T Y , ∇LY ) ∧ γ1 (T ).

iii) There exists C > 0 such that for ε ∈ (0, 1/T ′ ], εT ′ ≤ T ≤ 1, Z 1/2 1/2 LY −1 E −1 −1 T (2.93) γΩ (T ε ) ∧ chg (LY , ∇ ) ∧ chg (E, ∇ ) ≤ C. ε βg (T ε , ε) + Zg

iv) There exist δ ∈ (0, 1], C > 0 such that, for ε ∈ (0, 1], T ≥ 1, ε−1 |βgT (T ε−1 , ε)| ≤

(2.94)

C T 1+δ

.

Now using the theorems above, we can prove Lemma 2.16. By (2.81), we know that (2.95) Z Z A u βg (T0 , u)du −

T0

T′

ε

βgT (T, A)dT

−

Z

A

ε

βgu (T ′ , u)du

+

Z

T0

T′

βgT (T, ε)dT = I1 + I2 + I3 + I4

is an exact form. We take the limits A → ∞, T0 → ∞ and then ε → 0 in the indicated order. Let Ijk , j = 1, 2, 3, 4, k = 1, 2, 3 denote the value of the part Ij after the kth limit. By [17, §22, Theorem 17], dΩ(B) is closed under uniformly convergence on B. Thus, 4 X

(2.96)

j=1

Ij3 ≡ 0 mod d Ω∗ (B, C).

Since the definition of the equivariant eta form does not depend on the cut-off function in the definition, from (2.43), we obtain that modulo exact forms, b X )). I33 = η˜g (FZ,T ′ , T ′ (1⊗A

(2.97)

Furthermore, by Theorem 2.22, we get

I22 = I23 = 0.

(2.98) From Theorem 2.21, we have

I13 = 0.

(2.99)

Finally, using Theorem 2.24, we get Z LZ 3 g g ∇T Z Tdg (∇T Y , ∇LY ) ∧ ηeg (FX , AX ) + FLI , ∇T Y,T X , ∇LZ I4 = − (2.100) T′ , ∇ Yg

as follows: We write

Z

(2.101)

+∞

T′

βgT (T, ε)dT

=

Z

+∞ εT ′

ε−1 βgT (T ε−1 , ε)dT.

Convergence of the integrals above is granted by (2.91). Using (2.92), (2.94) and Proposition 2.23, we get Z +∞ Z Z +∞ TY LY −1 T −1 (2.102) γ1 (T )dT Tdg (∇ , ∇ ) ∧ lim ε βg (T ε , ε)dT = ε→0 1

Yg

1


29

and (2.103)

lim

Z

1

ε→0 εT ′

Z ε−1 βgT (T ε−1 , ε)dT +

Zg

1/2 1/2 γΩ (T ε−1 ) ∧ chg (LY , ∇LY ) ∧ chg (E, ∇E ) dT =

Z

Yg

Tdg (∇T Y , ∇LY ) ∧

Z

1

γ1 (T )dT. 0

The remaining part of the integral yields by (2.93) Z

1

Z

1/2

1/2

(2.104) lim γΩ (T ε−1 ) ∧ chg (LY , ∇LY ) ∧ chg (E, ∇E )dT ε ε→0 εT ′ Zg Z Z +∞ 1/2 1/2 LZ g g ∇T Z , ∇T Y,T X , ∇LZ . γΩ (T ) ∧ chg (LY , ∇LY ) ∧ chg (E, ∇E )dT = −FLI = T′ , ∇ Zg

−1

T′

These four equations for Ik3 , k = 1, 2, 3, 4, and (2.50) imply Lemma 2.16.

2.7. Proofs of Theorem 2.21-2.24. Since ker(D(FX ) + AX ) = 0, the proofs of Theorem 2.21-2.24 in our case are much easier. We only need to replace D X and DTZ somewhere in [23] b AX and take care in the local index computation in the by D(FX ) + AX and D(FZ,T ) + 1⊗T proof of Theorem 2.24 ii). In this subsection, we sketch the local index part here. Set ∂(Bε2 ,T ′ + εT ′ χ(εT ′ )AX ) ′ 2 −1 Bε,T /ε = (Bε2 ,T /ε + T χ(T )AX ) + ε dT ∧ (2.105) . ′ ∂T ′ −1 T =T ε

By the definition of

βgT (T, ε),

we have

n odT f exp(−B ′ ε−1 βgT (T /ε, ε) = ψS Tr[g )] . ε,T /ε

(2.106)

Let SX be the tensor in (2.19) with respect to πX . Let {ei }, {fp } and {gα } be the locally H } and {g H } be the corresponding horizontal orthonormal basis of T X, T Y and T B and {fp,1 α,3 lifts. Precisely, by (2.23), we have ′ ′ −1 ∂(Bε2 ,T ′ + εT χ(εT )AX ) = D X + χ(T )AX + T χ′ (T )AX (2.107) ε ′ ′ ∂T −1 T =T ε 1 H H H H (hε2 [fp,1 , fq,1 ], ei ic(ei )c(fp,1 )c(fq,1 ) − 8T 2 H H H H H ], ei ic(ei )gα ∧ g β ∧). + 4εhSX (gα,3 )ei , fp,1 ic(ei )c(fp,1 )g3α ∧ +h[gα,3 , gβ,3 As in (2.105), we set (2.108)

BT′′ 2 |V g

= (BX,T 2 |V g

Then by (2.89), we have (2.109)

∂(BX,T 2 + T χ(T )AX ) + T χ(T )AX ) + dT ∧ g. ∂T V 2

odT n f exp(−B ′′ 2 |V g )] γ1 (T ) = ψV g Tr[g . T

−1 (V g ) and As the same process in [23, Section 6], we could localize the problem near πX ′ define the operator Bε,T /ε to a neighborhood of {0} × Xy0 in Ty0 Y × Xy0 .

Let dV , dW be the distance functions on V , W associated to gT V , gT W . Let InjV , InjW be the injective radius of V , W . In the sequel, we assume that given 0 < α < α0 < inf{InjV , InjW }

30

BO LIU

are chosen small enough so that if y ∈ V , dV (g−1 y, y) ≤ α, then dV (y, V g ) ≤ 41 α0 , and if z ∈ W , dW (g−1 z, z) ≤ α, then dW (z, W g ) ≤ 14 α0 . Let ρ : Ty0 Y → [0, 1] be a smooth function such that 1, |U | ≤ α0 /4; (2.110) ρ(U ) = 0, |U | ≥ α0 /2. Let ∆T Y be the ordinary Laplacian operator on Ty0 Y . Set (2.111)

′ L1ε,T = (1 − ρ2 (U ))(−ε2 ∆T Y + T 2 (D X + AX )2y0 ) + ρ2 (U )Bε,T /ε .

For (U, x) ∈ NY g /Y,y0 × Xy0 , |U | < α0 /4, ε > 0, set (2.112)

(Sε s)(U, x) = s (U/ε, x) .

Put L2ε,T := Sε−1 L1ε,T Sε .

(2.113)

Let dim Ty0 Y g = l′ and dim NY g /Y,y0 = 2l′′ . Let {f1 , · · · , fl′ } be an orthonormal basis of Ty0 Y g and let {fl′ +1 , · · · , fl′ +2l′′ } be an orthonormal basis of NY g /Y,y0 . For α ∈ C(f p ∧ ′ ifp )1≤p≤l′ , let [α]max ∈ C be the coefficient of f 1 ∧ · · · ∧ f l in the expansion of α. Let Rε be a rescaling such that Rε (c(ei )) = c(ei ), (2.114)

f1p,H ∧ − ε if H , for 1 ≤ p ≤ l′ , p,1 ε H H Rε (c(fp,1 )) = c(fp,1 ), for l′ + 1 ≤ p ≤ l′ + 2l′′ .

H Rε (c(fp,1 )) =

Then Rε is a Clifford algebra homomorphism. Set (2.115)

L3ε,T = Rε (L2ε,T ).

Corresponding to [23, Lemma 3.4], from (2.106), (2.107) and (2.108), we have Lemma 2.25. When ε → 0, the limit L30,T = limε→0 L3ε,T 2 1 TY H 3 (2.116) L0,T |V g = − ∂p + hR |V g U, fp,1 i + 4

exists and 1 LY R |V g + BT′′ 2 |V g . 2

So all the computations here are the same as [23, Section 6]. 3. Equivariant differntial K-theory In this section, we assume that the the G-action on B is almost free (with finite stabilizers only). With this action, we construct an analytic model of equivariant differential K-theory and prove some properties using the results in Section 2. 3.1. Equivariant differential K-theory. In this subsection, we construct an analytic model of equivariant differential K-theory. When G = {e}, this construction is the same as that in [12]. Let E be a G-vector bundle over B. Then its restriction to B g is acted on fibrewisely by g for g ∈ G. So it decomposes as a direct sum of subbundles Eζ for each eigenvalue ζ of g. Set P 1 replace B by B × S 1 ) φg (E) := ζEζ . Then it induces a homomorphism (for KG (3.1)

∗ φg : KG (B) ⊗ C −→ [K ∗ (B g ) ⊗ C]CG (g) ,


31

where CG (g) is the centralizer of g in G. Let (g) be the conjugacy class of g ∈ G. For g, g ′ ∈ (g), there exists h ∈ G, such that g ′ = h−1 gh. Furthermore, the map ′

h : B g /CG (g′ ) → B g /CG (g)

(3.2)

′

′

is a homeomorphism . So [K ∗ (B g ) ⊗ C]CG (g) ≃ [K ∗ (B g ) ⊗ C]CG (g ) . By [1, Corollary 3.13], we know that the additive decomposition M ∗ (3.3) φ = ⊕(g),g∈G φg : KG (B) ⊗ C → [K ∗ (B g ) ⊗ C]CG (g) (g),g∈G

is an isomorphism, where (g) ranges over the conjugacy classes of G. Since B, G are compact and the group action is with finite stabilizers, the direct sum in (3.3) only has finite terms and does not depend the choice of the element in (g). From (3.2), we also know that the map h∗ induces an isomorphism ′

We denote by (3.5)

′

h∗ : [Ω∗ (B g , C)]CG (g) → [Ω∗ (B g , C)]CG (g ) .

(3.4)

Ω∗deloc,G (B, C) :=

o M n [Ω∗ (B g , C)]CG (g)

(g),g∈G

the set of delocalized differential forms, where {·} denotes the isomorphic class in sense of (3.4). The definition above does not depend on the choice of g ∈ (g). It is easy to see that the exterior differential operator d preserves Ω∗deloc,G (B, C). We denote by the delocalized de Rham ∗ (B, C) the cohomology of the differential complex (Ω∗deloc,G (B, C), d). cohomology Hdeloc,G Then from (3.1) and (3.3), the equivariant Chern character isomorphism can be naturally defined by ≃

(3.6)

∗ ∗ chG : KG (B) ⊗ C −→ Hdeloc,G (B, C), M K 7→ {ch(φg (K))} . (g),g∈G

We note that ch(φg (K)) = chg (K) is CG (g)-invariant by the definition. Observe that the fixed-point set for g-action coincides with that for g−1 -action. Set (compare with [30, (1)]) (3.7)

∗ ∗ (B, C) : ∀g ∈ G, cg−1 = cg }. (B, R) = {c = ⊕(g),g∈G {cg } ∈ Hdeloc,G Hdeloc,G

Let Ω∗deloc,G (B, R) ⊂ Ω∗deloc,G (B, C) be the ring of forms ω = ⊕(g),g∈G {ωg }, such that ∀g ∈ ∗ G, ωg−1 = ωg . Then Hdeloc,G (B, R) is the cohomology of the differential complex (Ω∗deloc,G (B, R), d). From (3.6), we have the isomorphism (3.8)

≃

∗ ∗ (B, R). chG : KG (B) ⊗ R −→ Hdeloc,G

Definition 3.1. (compare with [12, Definition 2.4]) A cycle for an equivariant differential K-theory class over B is a pair (F, ρ), where F ∈ F∗G (B) and ρ ∈ Ω∗deloc,G (B, R)/Im d. The cycle (F, ρ) is called even (resp. odd) if F is even (resp. odd) and ρ ∈ Ωodd deloc,G (B, R)/Im d ′ , ρ′ ) are called isomorphic if F and (resp. ρ ∈ Ωeven (B, R)/Im d). Two cycles (F, ρ) and (F deloc,G ′ c 1G (B)) denote the set of isomorphism c 0G (B) (resp. IC F are isomorphic and ρ = ρ′ . Let IC classes of even (resp. odd) cycles over B with a natural abelian semi-group structure by (F, ρ) + (F ′ , ρ′ ) = (F + F ′ , ρ + ρ′ ).

32

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For F ∈ F∗G (B), we assume that there exists a perturbation operator A with respect to D(F). For any g ∈ G, by Definition 2.11, the equivariant eta form restricted on the fixed point set of g is CG (g)-invariant, that is, ηeg (F|B g , A|B g ) ∈ [Ω∗ (B g , C)]CG (g) . Let h∗ be map in (3.4). Since the perturbation operator A is equivariant, from Definition 2.11, we have (3.9)

ηeg′ (F|B g′ , A|B g′ ) = h∗ ηeg (F|B g , A|B g ).

Since ηeg−1 (F|B g , A|B g ) = ηeg (F|B g , A|B g ), the following definition is well-defined.

Definition 3.2. The delocalized eta form ηeG (F, A) is defined by M (3.10) ηeG (F, A) = {e ηg (F|B g , A|B g )} ∈ Ω∗deloc,G (B, R). (g),g∈G

By the same process, we can define M (3.11) FLIG (F) = {FLIg (F)} ∈ Ω∗deloc,G (B, R). (g),g∈G

Let F ∈ F∗G (B) and A be a perturbation operator with respect to D(F). Then by Definition 1.3, there exists a perturbation operator Aop with respect to D(F op ) such that (3.12)

ηeG (F op , Aop ) = −e ηG (F, A).

Let F, F ′ ∈ F∗G (B), A, A′ be perturbation operators with respect to D(F), D(F ′ ). By Definition 2.11, we have (3.13)

ηeG (F + F ′ , A ⊔B A′ ) = ηeG (F, A) + ηeG (F ′ , A′ ).

As in [12, page 19], from Remark 2.4 , we know that for any F ∈ F∗G (B), there exists a perturbation operator A with respect to D(F + F op ) such that (3.14)

ηeG (F + F op , A) = 0.

Definition 3.3. (compare with [12, Definition 2.10]) We call two cycles (F, ρ) and (F ′ , ρ′ ) paired if ind(D(F)) = ind(D(F ′ )), and there exists a perturbation operator A with respect ′ to D(F + F op ) such that (3.15)

ρ − ρ′ = ηeG (F + F

′ op

, A).

Let ∼ denote the equivalence relation generated by the relation ”paired”. From (3.12), (3.13) and (3.14), we have Lemma 3.4. (compare with [12, Lemma 2.11, 2.12]) The relation ”paired” is symmetric, rec ∗ (B). flexive and compatible with the semigroup structure on IC G

Definition 3.5. (compare with [12, Definition 2.14]) The equivariant differential K-theory b 0 (B) (resp. K b 1 (B)) is the group completion of the abelian semigroup IC c even (B)/ ∼ (resp. K G G G odd c IC (B)/ ∼). G

Remark 3.6. Note that in [30, Section 3.3], the author constructed a geometric model of equivariant differential K 0 theory under the finite group action. From (2.45) and the explanab 0 in that paper is tions after it, we could easily see that when G is finite, the definition of K G isomorphic to that of ours.


33

b ∗ (B) the corresponding class in equivariant c ∗G (B), we denote by [F, ρ] ∈ K If (F, ρ) ∈ IC G b i (B), i = 0, 1, we have differential K-theory. From (3.12)-(3.14), for any [F, ρ], [F ′ , ρ′ ] ∈ K G [F, ρ] = [F + F

(3.16)

′ op

, ρ − ρ′ ] + [F ′ , ρ′ ].

b ∗ (B) can be represented in the form [F, ρ]. Furthermore, we have So every element of K G −[F, ρ] = [F op , −ρ]. 3.2. Push-forward map. In this subsection, we construct a well-defined push-forward map in equivariant differential K-theory and prove the functoriality of it using the theorems in Section 2. This solves a question proposed in [12] when G = {e}. We use the notations in Section 1.4. Let πY : V → B be an equivariant submersion of closed smooth G-manifolds with closed Spinc fiber Y . We assume that the G-action on B is almost free. Thus, so is the action on V . For g ∈ G, the fixed point set V g is the total space of the fiber bundle πY |V g : V g → B g with Y g . Since the pull back isomorphism h∗ in (3.4) commutes with the integral along the fiber, for α = ⊕(g),g∈G {αg } ∈ Ω∗deloc,G (V, R), the integral (3.17)

Z

α :=

Y,G

M Z

(g),g∈G

Y

g

αg

∈ Ω∗deloc,G (B, R)

does not depend on g ∈ (g). So it defines an integral map Z (3.18) : Ω∗deloc,G (V, R) → Ω∗deloc,G (B, R). Y,G

b∗ (πY ) of equivariant geometric data obY = (TπH V, gT Y , ∇LY , σY ), where Consider the set O G Y odd σY ∈ Ωdeloc,G (V )/Imd. Let M (3.19) Tdg (∇T Y , ∇LY ) ∈ Ω∗deloc,G (V, R). TdG (∇T Y , ∇LY ) = (g),g∈G

′ ′ ′ b∗ (πY ) be another equivariant tuple with the same equiLet ob′Y = (TπH V, g T Y , ∇ LY , σY′ ) ∈ O G Y variant K-orientation in Definition 1.7. As in (2.62), from [27, Theorem B.5.4], we can construct f G (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) ∈ Ωodd (V )/Imd such that the Chern-Simons form Td deloc,G

(3.20)

f G (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) = TdG (∇′ T Y , ∇′ LY ) − TdG (∇T Y , ∇LY ). d Td

We introduce a relation obY ∼ ob′Y as in [12]. Two equivariant tuples obY , ob′Y are related if and only if

(3.21)

′

′

f G (∇T Y , ∇LY , ∇ T Y , ∇ LY ), σY′ − σY = Td

where we mark the objects associated with the second tuple by ′ .

Definition 3.7. (compare with [12, Definition 3.5]) The set of equivariant differential Kb∗ (πY ))/ ∼. orientations is the set of equivalence classes O G b ∗ (V ) → K b ∗ (B) for a We now start with the construction of the push-forward map π bY ! : K G G given equivariant differential K-orientation which extends Theorem 1.8 to the differential case.

34

BO LIU

b ∗ (V ), let FZ be the equivariant geometric family defined in (1.25). We define For [FX , ρ] ∈ K G Z g G ∇T Z , ∇LZ , ∇T Y,T X , ∇LZ TdG (∇T Y , ∇LY ) ∧ ρ − FLI (3.22) π bY !([FX , ρ]) = FZ , Y,G

+

Z

Y,G

∗ bG σY ∧ (FLIG (FX ) − dρ) ∈ K (B),

T Z , ∇LZ , ∇T Y,T X , ∇LZ ∈ Ω∗ g g G ∇T Z , ∇LZ , ∇T Y,T X , ∇LZ = L where FLI (g),g∈G FLIg ∇ deloc,G (B, R).

b ∗ (V ) → K b ∗ (B) in (3.22) Theorem 3.8. (compare with [12, Lemma 3.14]) The map π bY ! : K G G is well-defined. ′ , ρ′ ) be two cycles over V . By (3.22), we have Proof. Let (FX , ρ), (FX

(3.23)

′

′ π bY !(FX , ρ) − π bY !(FX , ρ′ ) = π bY !(FX ⊔ FXop , ρ − ρ′ ).

′ , ρ′ ), there exists a perturbation operator A, such that If (FX , ρ) is paired with (FX ′

ρ − ρ′ = ηeG (FX ⊔ FXop , A).

(3.24)

So we only need to prove that if there exists a perturbation operator AX with respect to D(FX ), b ∗ (B). π bY !([FX , ηeG (FX , AX )]) = 0 ∈ K G From (3.22), we have Z TdG (∇T Y , ∇LY ) ∧ ηeG (FX , AX )) (3.25) π bY !([FX , ηeG (FX , AX )]) = FZ , Y,G

gG ∇ − FLI

TZ

,∇

LZ

,∇

T Y,T X

,∇

LZ

+

Z

Y,G

σY ∧ (FLIG (FX ) − de ηG (FX , AX )) .

From Proposition 2.3 (iii) and Lemma 2.15, there exists a perturbation operator AZ with ∗ (B) such respect to D(FZ ). By (2.44), (3.13), (3.17) and Theorem 2.18, there exists x ∈ KG that (3.26)

π bY !(FX , ηeG (FX , AX )) = [FZ , ηeG (FZ , AZ ) − chG (x)] .

From Proposition 2.7, there exist F ∈ F∗G (B) and equivariant spectral sections P , Q with respect to D(F), such that [P − Q] = x. Let AP , AQ be the perturbation operators associated with P , Q. From Theorem 2.17, we have (3.27)

chG (x) = ηeG (F, AP ) − ηeG (F, AQ ).

By (3.12), (3.26), (3.27) and Definition 3.3, we have h i (3.28) π bY !(FX , ηeG (FX , AX )) = FZ , ηeG (FZ , AZ ) − ηeG (F, AP ) − ηeG (F op , Aop ) Q

b ∗ (B). = [F + F op , 0] = [F, 0] − [F, 0] = 0 ∈ K G

Then from Theorem 1.8, we complete the proof of Theorem 3.8.

Here our construction of π bY ! involve an explicit choice of a representative obY = (TπHY V, gT Y , , σY ) of the equivariant differential K-orientation. In fact, it does not depend on the choice.

∇ LY

b ∗ (V ) → K b ∗ (B) Lemma 3.9. (compare with [12, Lemma 3.17]) The homomorphism π bY ! : K G G only depend on the equivariant differential K-orientation.

EQUIVARIANT ETA FORMS AND EQUIVARIANT DIFFERENTIAL K-THEORY ′

′

35

′

V, g T Y , ∇ LY , σY′ ) be two representatives of Proof. Let obY = (TπHY V, gT Y , ∇LY , σY ), ob′Y = (TπH Y an equivariant differential K-orientation. We will mark the objects associated with the second representative by ′ . From (2.63), we could get (3.29) g G (∇T Y,T X , ∇LZ , ∇′ T Y,T X , ∇′ LZ ) = FLI

Z

Y,G

Then from (3.20), (3.21) and (3.29), we have

f G (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) ∧ FLIG (FX ). Td

bY !([FX , ρ]) = [FZ′ + FZop , (3.30) π bY′ !([FX , ρ]) − π Z ′ ′ g G ∇′ T Z , ∇′ LZ , ∇′ T Y,T X , ∇′ LZ TdG (∇ T Y , ∇ LY ) − TdG (∇T Y , ∇LY ) ∧ ρ − FLI Y,G

g G ∇T Z , ∇LZ , ∇T Y,T X , ∇LZ + +FLI

−

Z

Y,G

(σY′

− σY ) ∧ (FLIG (FX ) − dρ)

f G (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) ∧ ρ d Td Y,G Z ′ ′ f G (∇T Y , ∇LY , ∇′ T Y , ∇′ LY ) ∧ FLIG (FX ) f G (∇T Y , ∇LY , ∇ T Y , ∇ LY ) ∧ dρ + Td Td = FZ′ + FZop ,

Y,G

Z

Z

Y,G

i g G ∇T Y,T X , ∇LZ , ∇′ T Y,T X , ∇′ LZ g G ∇T Z , ∇LZ , ∇′ T Z , ∇′ LZ − FLI + FLI

g G FZ , F ′ ]. = [FZ′ + FZop , FLI Z

By Proposition 2.3 (iii) and Lemma 2.15, there exists a perturbation operator A with respect ∗ (B) such that to D(FZ′ + FZop ). By Theorem 2.17 and (3.14), there exists x ∈ KG g G FZ + F op , FZ′ + F op = −e g G FZ , FZ′ = FLI (3.31) ηG (FZ′ + FZop , A) + chG (x). FLI Z Z Following the same process in (3.26)-(3.28), we have π bY′ !([FX , ρ]) = π bY !([FX , ρ]). The proof of Lemma 3.9 is complete.

We now discuss the functoriality of the push-forward maps with respect to the composition of fiber bundles. Let πY : V → B with fiber Y be as in the above subsection together with a representative of an equivariant differential K-orientation obY = (TπHY V, gT Y , ∇LY , σY ). Let πU : B → S be another equivariant proper submersion with closed fibers U together with a representative of an equivariant differential K-orientation obU = (TπHU B, gT U , ∇LU , σU ). Let πA := πU ◦ πY : V → S be the composition of two submersions with fibers A. Let TπHA V be a horizontal subbundle associated with πA . We assume that TπHA V ⊂ TπHY V . Set gT A = πY∗ g T U ⊕ gT Y , ∇LA = πY∗ ∇LU ⊗ ∇LY . Definition 3.10. (compare with [12, Definition 3.21]) We define the composite obA = obU ◦ obY of the representatives of equivariant differential K-orientations of πY and πU by

(3.32)

where

obA := (TπHA V, gT A , ∇LA , σA ),

(3.33) σA := σY ∧ πY∗ TdG (∇T U , ∇LU ) + TdG (∇T Y , ∇LY ) ∧ πY∗ σU

f G (∇T A , ∇LA , ∇T U,T Y , ∇LA ) − dσY ∧ π ∗ σU . + Td Y

36

BO LIU

Theorem 3.11. (compare with [12, Theorem 3.23]) We have the equality of homomorphisms b ∗ (V ) → K b ∗ (S) K G G π bA ! = π bU ! ◦ π bY !.

(3.34)

Proof. The topological part of Theorem 3.11 is just Theorem 1.9 and the differential part follows from a direct calculation using (3.22) and (3.33). 3.3. Cup product. In this subsection, we construct the cup product of equivariant differential K-theory in our model as in [12, 14] and prove the desired properties. b i (B) and [F ′ , ρ′ ] ∈ K b ∗ (B), where i = 0, 1. We define (compare with [12, Let [F, ρ] ∈ K G G Definition 4.1]) (3.35)

[F, ρ] ∪ [F ′ , ρ′ ] := [F ∪ F ′ , (−1)i FLIG (F) ∧ ρ′ + ρ ∧ FLIG (F ′ ) − (−1)i dρ ∧ ρ′ ].

Proposition 3.12. (compare with [12, Proposition 4.2, 4.5]) (i) The product is well defined. It b ∗ (B) into a functor from closed smooth G-manifolds to unital graded commutative turns B 7→ K G

rings. The unit is simply given by [F, 0], where F is the equivariant geometric family in Example 1.4 a) such that E+ is 1 dimensional trivial representation and E− = 0. (ii) The product is associative. (iii) Let πU : B → S be an equivariant proper submersion with an equivariant differential b ∗ (B) and y ∈ K b ∗ (S), we have K-orientation. For x ∈ K G G π bU !(πU∗ y ∪ x) = y ∪ π bU !(x).

(3.36)

Proof. The product is obviously biadditive. Let f : B1 → B2 be a G-equivariant smooth map. Let F ∈ F∗G (B2 ). Then we can naturally define f ∗ F ∈ F∗G (B1 ). We only need to be careful with the the pull back of the horizontal subbundle. Let F be the natural map from f ∗ W to W . The new horizontal subbundle TfH∗ π (f ∗ W ) is chosen by the condition that dF (TfH∗ π (f ∗ W )) ⊆ TπH (W ). (We note that the chosen of the new horizontal subbundle is not unique. But it is unique in the differential K-theory level.) Let A be a perturbation operator with respect to D(F). Then f ∗ A is a perturbation operator with respect to D(f ∗ F). Moreover, from Definition 2.11, we have ηeG (f ∗ F, f ∗ A) = f ∗ ηeG (F, A).

(3.37)

b ∗ (B2 ) → K b ∗ (B1 ). Evidently, From Definition 3.3, we can get a well defined pullback map f ∗ : K G G ′ ∗ ′ IdB = IdKb G (B) . Let f : B0 → B1 be another equivariant smooth map. We could get f ∗ f ∗ = b ∗ (B2 ) → K b ∗ (B0 ). It is obvious that the product is natural with respect to (f ◦ f ′ )∗ : K G

G

pull-backs. From Theorem 2.19 and a direct calculation, we could get the product is compatible with the equivalent relation in differential K-theory. Other properties are the direct extension of the discussions in [12, page 47-50]. b G is Theorem 3.13. (compare with [12, Section 3,4]) The equivariant differential K-theory K b a functor B → KG (B) from the category of closed smooth G-manifolds with almost free action to unital Z2 -graded commutative rings together with well-defined transformations b ∗ (B) → Ω∗ (1) R : K G deloc,G,cl (B, R) (curvature); ∗ ∗ b (B) → K (B) (underlying KG group); (2) I : K G G b G (X) (action of forms), (3) a : Ω∗ (B, R)/Im d → K deloc,G

where Ω∗deloc,G,cl (B, R) denote the set of closed delocalized differential forms, such that (1) The following diagram commutes


I

b ∗ (B) K G

37

∗ (B) KG

chG

R

Ω∗deloc,G,cl (B, R)

de Rham

∗ Hdeloc,G (B, R).

(2) R ◦ a = d.

(3.38)

(3) a is of degree 1. b ∗ (B) and α ∈ Ω∗ (4) For x, y ∈ K G deloc,G (B, R)/Imd, we have

(3.39)

R(x ∪ y) = R(x) ∧ R(y),

I(x ∪ y) = I(x) ∪ I(y),

a(α) ∪ x = a(α ∧ R(x)).

(5) The following sequence is exact: (3.40)

chG a I ∗ ∗−1 b∗ (B) −→ Ω∗−1 KG deloc,G (B, R)/Im d −→ KG (B) −→ KG (B) −→ 0.

Proof. We define the natural transformation (3.41) by (3.42)

∗ ∗ bG I:K (B) → KG (B)

I([F, ρ]) := ind(D(F)).

From Definition 3.3, the transformation I is well defined. Let a be a parity-reversing natural transformation (3.43) by

even/odd

1/0

b (B) a : Ωdeloc,G (B, R)/Im d → K G a(ρ) := [∅, −ρ],

(3.44) where ∅ is the empty geometric family. We define a transformation (3.45) by (3.46)

∗

c G (B) → Ω∗ R : IC deloc,G,cl (B, R) R((F, ρ)) := FLIG (F) − dρ.

If (F ′ , ρ′ ) is paired with (F, ρ), there exists a perturbation operator A with respect to D(F + ′ ′ F op ), such that ρ − ρ′ = ηeG (F + F op , A). From (2.44) and (3.15), we have

(3.47) R((F, ρ)) = FLIG (F) − dρ = FLIG (F) − dρ′ − de ηG (F + F

′ op

, A)

= FLIG (F) − dρ′ − FLIG (F) + FLIG (F ′ ) = R((F ′ , ρ′ )).

b ∗ (B) → Ω∗ c ∗ (B)/ ∼ and finally to the map R : K Since R is additive, it descends to IC G G deloc,G,cl (B, R). Let f : B1 → B2 be a G-equivariant smooth map. It follows from FLIG (f ∗ F) = f ∗ FLIG (F) that R is natural. From (3.44) and (3.46), we have (3.48)

R ◦ a = d.

By (2.35), the diagram commutes. The formulas in (3.39) follow from straight calculations using the definitions. At last, we prove the exactness of the sequence (3.40).

38

BO LIU

The surjectivity of I follows from Proposition 1.5. b ∗ (B). It is obvious that I ◦ a = 0. For a cycle (F, ρ), if Next, we show the exactness at K G I([F, ρ]) = 0, we have ind(D(F)) = 0. By Example 1.4 b), we could take F such that at least one component of the fiber has the nonzero dimension. So there exists a perturbation operator A with respect to D(F) from Proposition 2.3. By (3.15), we have [F, ρ] = a(e ηG (F, A) − ρ).

(3.49)

Finally, We prove the exactness at Ω∗−1 deloc,G,cl (B, R)/Im d. Following the same process in ∗ (B), (3.26)-(3.28), for any x ∈ KG a ◦ chG (x) = (∅, η˜G (F, AQ ) − η˜G (F, AP )) = [F, 0] − [F, 0] = 0.

(3.50)

If a(ρ) = 0, for any equivariant geometric family F with a perturbation operator A with respect to D(F), by Definition 3.3 and (3.49), we have [F, η˜G (F, A) − ρ] = a(ρ) = 0 = [F, η˜G (F, A)].

(3.51)

So by Definition 3.5, there exists another cycle (F ′ , ρ′ ), such that (F + F ′ , ρ′ + η˜G (F, A) − ρ) ∼ (F +F ′ , ρ′ +˜ ηG (F, A)). Since ∼ is generated by ”paired”, we have the cycles {(Fi , ρi )}0≤i≤r such that for any 1 ≤ i ≤ r, (Fi , ρi ) is paired with (Fi−1 , ρi−1 ), (F0 , ρ0 ) = (F + F ′ , ρ′ + η˜G (F, A)− ρ) and (Fr , ρr ) = (F + F ′ , ρ′ + η˜G (F, A)). By Definition 3.3, for any 1 ≤ i ≤ r, there exists a perturbation operator Ai with respect to D(Fi−1 +Fiop ) such that ρi−1 −ρi = η˜G (Fi−1 +Fiop , Ai ). Let A′i (0 ≤ i ≤ r) be the perturbation operator with respect to D(Fi + Fiop ) taken in (3.14). ( Therefore, by Theorem 2.17, (3.13) and (3.14), there exists x ∈ KG B), such that (3.52)

−ρ=

r X i=1

(ρi−1 − ρi ) = η˜G (F0 + F1op + · · · + Fr−1 + Frop , A1 ⊔B · · · ⊔B Ar ) op = η˜G (F0 + Frop + · · · + Fr−1 + Fr−1 , A1 ⊔B · · · ⊔B Ar )

op − η˜G (F0 + Frop + · · · + Fr−1 + Fr−1 , A′0 ⊔B · · · ⊔B A′r−1 ) = chG (x).

The proof of Theorem 3.13 is complete.

The direct extension of [12, Proposition 3.19 and Lemma 3.20] show that the pullback map and the exact sequence (3.40) are compatible with the push-forward maps. Remark 3.14. If the group G is trivial, all the models of differential K-theory are isomorphic (see e.g. [13]). For equivariant case, the uniqueness is an open question. 3.4. Differential K-theory for orbifolds. Let X be a closed orbifold (effective orbifold in some literatures). There exist a closed smooth manifold B and a compact Lie group G such that X is diffeomorphic to a quotient for a smooth, effective, and almost free G-action on B (see [1, Theorem 1.23]). 0 (X ) be the orbifold K-group of the compact orbifold X defined as the Grothendieck Let Korb ring of the equivalent classes of orbifold vector bundles over X . Since X is an orbifold, X × S 1 is an orbifold. Moreover, i : X → X × S 1 is a morphism in the category of orbifolds. As in 1 (X ) = ker(i∗ : K 0 (X × S 1 ) → K 0 (X )). (1.12), we define the orbifold K 1 group Korb orb orb Let p : B → B/G be the projection. Then from [1, Proposition 3.6], it induces an isomor∗ (X ) → K ∗ (B). phism p∗ : Korb G So we can consider the orbifold K-theory as a special case of equivariant K-theory. Note that if the orbifold X can be presented in two differential ways as a quotient, say B ′ /G′ ≃ X ≃ B/G, ∗ (B ′ ) ≃ K ∗ (X ) ≃ K ∗ (B). Furthermore, from the definition of the proposition shows that KG ′ G orb


39

the differential structure on orbifolds, we know that Ω∗deloc,G (B, R)/Imd ≃ Ω∗deloc,G′ (B ′ , R)/Imd. From the exact sequence in (3.40) and five lemma, we have (3.53)

∗ bG b ∗ ′ (B ′ ) ≃ K (B). K G

Therefore, this model of equivariant differential K-theory for almost free action could be regarded as a model of differential K-theory for orbifolds. Acknowledgements I would like to thank Prof. Xiaonan Ma for many discussions. I would also like to thank Shu Shen and Guoyuan Chen for the conversations. I would like to thank the Mathematical Institute of Humboldt-Universit¨ at zu Berlin, especially Prof. J. Br¨ uning for financial support and hospitality. This research has been financially supported by Collaborative Research Centre ”Space Time - Matte” (SFB 647) References [1] A. Adem, J. Leida, and Y. Ruan. Orbifolds and stringy topology, volume 171 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2007. ´ [2] M. F. Atiyah and I. M. Singer. Index theory for skew-adjoint Fredholm operators. Inst. Hautes Etudes Sci. Publ. Math., (37):5–26, 1969. [3] M. F. Atiyah and I. M. Singer. The index of elliptic operators. IV. Ann. of Math. (2), 93:119–138, 1971. [4] N. Berline, E. Getzler, and M. Vergne. Heat kernels and Dirac operators. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. [5] J.-M. Bismut. The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math., 83(1):91–151, 1985. [6] J.-M. Bismut and J. Cheeger. η-invariants and their adiabatic limits. J. Amer. Math. Soc., 2(1):33–70, 1989. [7] J.-M. Bismut and D. Freed. The analysis of elliptic families. I. Metrics and connections on determinant bundles. Comm. Math. Phys., 106(1):159–176, 1986. [8] J.-M. Bismut and D. Freed. The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Comm. Math. Phys., 107(1):103–163, 1986. ´ [9] J.-M. Bismut and G. Lebeau. Complex immersions and Quillen metrics. Inst. Hautes Etudes Sci. Publ. Math., (74):ii+298 pp. (1992), 1991. [10] U. Bunke. Index theory, eta forms, and Deligne cohomology. Mem. Amer. Math. Soc., 198(928):vi+120, 2009. [11] U. Bunke and X. Ma. Index and secondary index theory for flat bundles with duality. In Aspects of boundary problems in analysis and geometry, volume 151 of Oper. Theory Adv. Appl., pages 265–341. Birkh¨ auser, Basel, 2004. [12] U. Bunke and T. Schick. Smooth K-theory. Astérisque, (328):45–135 (2010), 2009. [13] U. Bunke and T. Schick. Differential K-theory: a survey. In Global differential geometry, volume 17 of Springer Proc. Math., pages 303–357. Springer, Heidelberg, 2012. [14] U. Bunke and T. Schick. Differential orbifold K-theory. J. Noncommut. Geom., 7(4):1027–1104, 2013. [15] X. Dai. Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence. J. Amer. Math. Soc., 4(2):265–321, 1991. [16] X. Dai and W. Zhang. Higher spectral flow. J. Funct. Anal., 157(2):432–469, 1998. [17] G. de Rham. Variétés différentiables. Formes, courants, formes harmoniques. Hermann, Paris, 1973. Troisième édition revue et augmentée, Publications de l’Institut de Mathématique de l’Université de Nancago, III, Actualités Scientifiques et Industrielles, No. 1222b. [18] H. Fang. Equivariant spectral flow and a Lefschetz theorem on odd-dimensional Spin manifolds. Pacific J. Math., 220(2):299–312, 2005. [19] D. S. Freed and J. Lott. An index theorem in differential K-theory. Geom. Topol., 14(2):903–966, 2010. [20] M. J. Hopkins and I. M. Singer. Quadratic functions in geometry, topology, and M-theory. J. Differential Geom., 70(3):329–452, 2005. [21] H. B. Lawson and M.-L. Michelsohn. Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989.

40

BO LIU

[22] E. Leichtnam and P. Piazza. Dirac index classes and the noncommutative spectral flow. J. Funct. Anal., 200(2):348–400, 2003. [23] B. Liu. Functoriality of equivariant eta forms. arXiv:1505.04454. [24] X. Ma. Formes de torsion analytique et familles de submersions. I. Bull. Soc. Math. France, 127(4):541–621, 1999. [25] X. Ma. Submersions and equivariant Quillen metrics. Ann. Inst. Fourier (Grenoble), 50(5):1539–1588, 2000. [26] X. Ma. Functoriality of real analytic torsion forms. Israel J. Math., 131:1–50, 2002. [27] X. Ma and G. Marinescu. Holomorphic Morse inequalities and Bergman kernels, volume 254 of Progress in Mathematics. Birkh¨ auser Verlag, Basel, 2007. [28] R. B. Melrose and P. Piazza. Families of Dirac operators, boundaries and the b-calculus. J. Differential Geom., 46(1):99–180, 1997. [29] R. B. Melrose and P. Piazza. An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary. J. Differential Geom., 46(2):287–334, 1997. [30] M. L. Ortiz. Differential equivariant K-theory. ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–The University of Texas at Austin. [31] N. C. Phillips. Equivariant K-theory and freeness of group actions on C ∗ -algebras, volume 1274 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987. [32] D. Quillen. Superconnections and the Chern character. Topology, 24(1):89–95, 1985. ´ [33] G. Segal. Equivariant K-theory. Inst. Hautes Etudes Sci. Publ. Math., (34):129–151, 1968. [34] J. Simons and D. Sullivan. Structured vector bundles define differential K-theory. In Quanta of maths, volume 11 of Clay Math. Proc., pages 579–599. Amer. Math. Soc., Providence, RI, 2010. [35] R. J. Szabo and A. Valentino. Ramond-Ramond fields, fractional branes and orbifold differential K-theory. Comm. Math. Phys., 294(3):647–702, 2010. [36] E. Witten. D-branes and K-theory. J. High Energy Phys., (12):Paper 19, 41 pp. (electronic), 1998. ¨ t zu Berlin , Johann von Neumann - Haus, Rudower Mathematisches Institut, Humboldt-Universita Chaussee 25, 12489 Berlin, Deutschland E-mail address: boliumath@gmail.com